Result: Extended convergence analysis of the Scholtes-type regularization for cardinality-constrained optimization problems

Title:

Extended convergence analysis of the Scholtes-type regularization for cardinality-constrained optimization problems

Authors:

Sebastian Lämmel, Vladimir Shikhman

Source:

Mathematical Programming. 211:207-243

Publication Status:

Preprint

Publisher Information:

Springer Science and Business Media LLC, 2024.

Publication Year:

2024

Subject Terms:

index, nondegenerate T-stationarity, 0211 other engineering and technologies, Scholtes-type regularization method, 02 engineering and technology, cardinality-constrained optimization problem, Nonconvex programming, global optimization, 01 natural sciences, Optimization and Control (math.OC), FOS: Mathematics, 90C26, 49M20, Optimality conditions and duality in mathematical programming, 0101 mathematics, genericity, Mathematics - Optimization and Control

Document Type:

Academic journal Article

File Description:

application/xml

Language:

English

ISSN:

1436-4646
0025-5610

DOI:

10.1007/s10107-024-02082-3

DOI:

10.48550/arxiv.2212.14577

Access URL:

http://arxiv.org/abs/2212.14577
https://zbmath.org/8035685
https://doi.org/10.1007/s10107-024-02082-3

Rights:

CC BY

Accession Number:

edsair.doi.dedup.....37d617a76e6e5d1e02932fc6c4c1104c

Database:

OpenAIRE

Further Information

We extend the convergence analysis of the Scholtes-type regularization method for cardinality-constrained optimization problems. Its behavior is clarified in the vicinity of saddle points, and not just of minimizers as it has been done in the literature before. This becomes possible by using as an intermediate step the recently introduced regularized continuous reformulation of a cardinality-constrained optimization problem. We show that the Scholtes-type regularization method is well-defined locally around a nondegenerate T-stationary point of this regularized continuous reformulation. Moreover, the nondegenerate Karush–Kuhn–Tucker points of the corresponding Scholtes-type regularization converge to a T-stationary point having the same index, i.e. its topological type persists. As consequence, we conclude that the global structure of the Scholtes-type regularization essentially coincides with that of CCOP.

AN0184787735;3on01may.25;2025Apr30.03:17;v2.2.500

Extended convergence analysis of the Scholtes-type regularization for cardinality-constrained optimization problems

Keywords: Cardinality-constrained optimization problem; Scholtes-type regularization method; Nondegenerate T-stationarity; Index; Genericity; 90C26; 90C46; Mathematical Sciences Numerical and Computational Mathematics

Introduction

In nonconvex optimization Scholtes-type regularization methods became popular since the seminal paper [[1]]. Typically, nonsmooth constraints are relaxed by means of a parameter. Then, Karush–Kuhn–Tucker points of the induced nonlinear programs need to be computed. They are shown to converge towards some suitably defined stationary points of the original optimization problem as the regularization parameter tends to zero. Scholtes-type regularization methods for mathematical programs with complementarity (MPCC), vanishing (MPVC), switching (MPSC), and orthogonality type constrains (MPOC) were examined along these lines in the literature so far, see e.g. [[1], [3]–[4]] for further details, respectively.

In this paper, we study the Scholtes-type regularization method for the class of cardinality-constrained optimization problems:

Graph

with the feasible set given by equality, inequality, and cardinality constraints, where the so-called zero "norm" <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mfenced close="∥" open="∥"><mi>x</mi></mfenced><mn>0</mn></msub><mo>=</mo><mfenced close="|" open="|"><mfenced close="}" open="{"><mrow><mi>i</mi><mo>∈</mo><mrow><mo stretchy="false">{</mo><mn>1</mn><mo>,</mo><mo>...</mo><mo>,</mo><mi>n</mi><mo stretchy="false">}</mo></mrow><mspace width="0.277778em" /><mo stretchy="false">|</mo><mspace width="0.277778em" /></mrow><msub><mi>x</mi><mi>i</mi></msub><mo>≠</mo><mn>0</mn></mfenced></mfenced></mrow></math> is counting non-zero entries of x. Here, we assume that the objective function f, as well as the equality and inequality constraints <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>h</mi><mo>=</mo><mfenced close=")" open="("><msub><mi>h</mi><mi>p</mi></msub><mo>,</mo><mi>p</mi><mo>∈</mo><mi>P</mi></mfenced></mrow></math> , <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>g</mi><mo>=</mo><mfenced close=")" open="("><msub><mi>g</mi><mi>q</mi></msub><mo>,</mo><mi>q</mi><mo>∈</mo><mi>Q</mi></mfenced></mrow></math> are twice continuously differentiable, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>s</mi><mo>∈</mo><mo stretchy="false">{</mo><mn>1</mn><mo>,</mo><mo>...</mo><mo>,</mo><mi>n</mi><mo>-</mo><mn>1</mn><mo stretchy="false">}</mo></mrow></math> is an integer. In order to arrive at the Scholtes-type regularization, the so-called continuous reformulation of CCOP from [[5]] is helpful:

Graph

As pointed out there, <math xmlns="http://www.w3.org/1998/Math/MathML"><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></math> solves CCOP if and only if there exists a vector <math xmlns="http://www.w3.org/1998/Math/MathML"><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close=")" open="("><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mfenced></math> solves (1). In order to tackle (1) numerically, [[6]] suggests to regularize the orthogonality type constraints by using the Scholtes' idea, cf. [[1]]:

2 <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtable><mtr><mtd columnalign="right"><mstyle displaystyle="true" scriptlevel="0"><mrow><munder><mo movablelimits="true">min</mo><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow></munder><mspace width="0.166667em" /><mspace width="0.166667em" /><mi>f</mi><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mspace width="1em" /><mspace width="0.333333em" /><mtext>s.</mtext><mspace width="0.333333em" /><mspace width="0.166667em" /><mspace width="0.333333em" /><mtext>t.</mtext><mspace width="0.333333em" /><mspace width="0.166667em" /><mi>h</mi><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mo>=</mo></mrow></mstyle></mtd><mtd columnalign="left"><mstyle displaystyle="true" scriptlevel="0"><mrow><mrow /><mn>0</mn><mo>,</mo><mspace width="1em" /><mi>g</mi><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mo>≥</mo><mn>0</mn><mo>,</mo><mspace width="1em" /><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msub><mi>y</mi><mi>i</mi></msub><mo>≥</mo><mi>n</mi><mo>-</mo><mi>s</mi><mo>,</mo></mrow></mstyle></mtd></mtr><mtr><mtd columnalign="right"><mrow><mrow /><mo>-</mo><mi>t</mi><mo>≤</mo></mrow></mtd><mtd columnalign="left"><mrow><mrow /><msub><mi>x</mi><mi>i</mi></msub><msub><mi>y</mi><mi>i</mi></msub><mo>≤</mo><mi>t</mi><mo>,</mo><mspace width="1em" /><mn>0</mn><mo>≤</mo><msub><mi>y</mi><mi>i</mi></msub><mo>≤</mo><mn>1</mn><mo>,</mo><mspace width="1em" /><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo>...</mo><mo>,</mo><mi>n</mi><mo>,</mo></mrow></mtd></mtr></mtable></mrow></math>

Graph

where <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>t</mi><mo>></mo><mn>0</mn></mrow></math> . Further in [[7]], the authors prove that—under some suitable constraint qualification and second-order sufficient condition—the Scholtes-type regularization method is well-defined locally around a minimizer of (1). Moreover, the Karush–Kuhn–Tucker points of (2) converge to an S-stationary point of (1) whenever <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>t</mi><mo stretchy="false">→</mo><mn>0</mn></mrow></math> .

Our goal is to extend the convergence analysis of the Scholtes-type regularization method beyond the case of minimizers of (1), but also for all kinds of its saddle points. By doing so, we intend to relate the indices of nondegenerate Karush–Kuhn–Tucker points of the Scholtes-type regularization with those of T-stationary points of the regularized continuous reformulation. Here, nondegeneracy refers to some tailored versions of linear independence constraint qualification, strict complementarity and second-order regularity. Assuming nondegeneracy, Karush–Kuhn–Tucker points and T-stationary points can be classified according to their quadratic and T-index, respectively. The index encodes the local structure of the optimization problem under consideration in algebraic terms and its global structure in the sense of Morse theory, see [[8]]. We note that for our purpose we need to preliminarily regularize the continuous reformulation (1). The reason is that all T-stationary points of (1)—considered as an MPOC instance—turn out to be degenerate, cf. [[4]]. To overcome this obstacle, it has been suggested in [[9]] not only to linearly perturb the objective function in (1) with respect to y-variables, but also to additionally relax the upper bounds on them. As for our main results, the Scholtes-type regularization method proves to be well-defined locally around a nondegenerate T-stationary point of the regularized continuous reformulation. Moreover, the nondegenerate Karush–Kuhn–Tucker points of its Scholtes-type regularization converge to a T-stationary point having the same index. These results allow us to relate the x-variables of the Karush–Kuhn–Tucker points of the Scholtes-type regularization to the M-stationary points of CCOP directly.

We emphasize that the study of saddle points for the Scholtes-type regularization is not only valuable from the global optimization perspective, but also from the practical point of view. Indeed, since the Scholtes-type regularization falls into the scope of nonlinear programming, we can only hope to efficiently compute its Karush–Kuhn–Tucker points. This can be done e.g. by using Newton-type methods, which—as well known—do not in general converge towards minimizers. These Karush–Kuhn–Tucker points of the Scholtes-type regularization will thus appear to be saddle points of different kinds. Their convergence to the saddle points of the regularized continuous reformulation of CCOP and of CCOP itself has then to be addressed.

The article is organized as follows. In Sect. 2 we discuss some preliminary results on CCOP and its regularized continuous reformulation. Sect. 3 is devoted to the extended convergence analysis of its Scholtes-type regularization.

Our notation is standard. The cardinality of a finite set A is denoted by <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">|</mo><mi>A</mi><mo stretchy="false">|</mo></mrow></math> . The n-dimensional Euclidean space is denoted by <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mrow><mi mathvariant="double-struck">R</mi></mrow><mi>n</mi></msup></math> with the coordinate vectors <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>e</mi><mi>i</mi></msub><mo>,</mo><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo>...</mo><mo>,</mo><mi>n</mi></mrow></math> . The vector consisting of ones is denoted by e. Given a twice continuously differentiable function <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>f</mi><mo>:</mo><msup><mrow><mi mathvariant="double-struck">R</mi></mrow><mi>n</mi></msup><mo stretchy="false">→</mo><mi mathvariant="double-struck">R</mi></mrow></math> , <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="normal">∇</mi><mi>f</mi></mrow></math> denotes its gradient, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mi>D</mi><mn>2</mn></msup><mi>f</mi></mrow></math> stands for its Hessian.

Preliminaries

We start with the notion of nondegenerate stationarity for CCOP as described in [[10]]. For that, we use the index set of active inequality constraints and the index set of vanishing x-variables, i.e.

Graph

Let us introduce the CCOP-tailored linear independence constraint qualification.

Definition 1

(CC-LICQ, see [[11]]) We say that a feasible point <math xmlns="http://www.w3.org/1998/Math/MathML"><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></math> of CCOP satisfies the cardinality-constrained linear independence constraint qualification (CC-LICQ) if the following gradients are linearly independent:

Graph

It was shown in [[10]] that the topologically relevant stationary concept for CCOP is M-stationarity, namely in the sense of the Morse theory.

Definition 2

(M-stationarity, see [[5]]) A CCOP feasible point <math xmlns="http://www.w3.org/1998/Math/MathML"><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></math> is called M-stationary if there exist multipliers

Graph

such that the following conditions hold:

3 <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtable><mtr><mtd columnalign="right"><mrow><mi mathvariant="normal">∇</mi><mi>f</mi><mo stretchy="false">(</mo><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo stretchy="false">)</mo><mo>=</mo></mrow></mtd><mtd columnalign="left"><mrow><mrow /><munder><mo movablelimits="false">∑</mo><mrow><mi>p</mi><mo>∈</mo><mi>P</mi></mrow></munder><msub><mover accent="true"><mrow><mi>λ</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mi>p</mi></msub><mi mathvariant="normal">∇</mi><msub><mi>h</mi><mi>p</mi></msub><mrow><mo stretchy="false">(</mo><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo stretchy="false">)</mo></mrow><mo>+</mo><munder><mo movablelimits="false">∑</mo><mrow><mi>q</mi><mo>∈</mo><msub><mi>Q</mi><mn>0</mn></msub><mrow><mo stretchy="false">(</mo><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo stretchy="false">)</mo></mrow></mrow></munder><msub><mover accent="true"><mrow><mi>μ</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mi>q</mi></msub><mi mathvariant="normal">∇</mi><msub><mi>g</mi><mi>q</mi></msub><mrow><mo stretchy="false">(</mo><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo stretchy="false">)</mo></mrow><mo>+</mo><munder><mo movablelimits="false">∑</mo><mrow><mi>i</mi><mo>∈</mo><msub><mi>I</mi><mn>0</mn></msub><mrow><mo stretchy="false">(</mo><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo stretchy="false">)</mo></mrow></mrow></munder><msub><mover accent="true"><mrow><mi>γ</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mi>i</mi></msub><msub><mi>e</mi><mi>i</mi></msub><mo>,</mo></mrow></mtd></mtr></mtable></mrow></math>

Graph

4 <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtable><mtr><mtd columnalign="right"><mrow><msub><mover accent="true"><mrow><mi>μ</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mi>q</mi></msub><mo>≥</mo></mrow></mtd><mtd columnalign="left"><mrow><mrow /><mn>0</mn><mspace width="0.333333em" /><mtext>for</mtext><mspace width="0.333333em" /><mspace width="0.333333em" /><mtext>all</mtext><mspace width="0.333333em" /><mi>q</mi><mo>∈</mo><msub><mi>Q</mi><mn>0</mn></msub><mrow><mo stretchy="false">(</mo><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo stretchy="false">)</mo></mrow><mo>.</mo></mrow></mtd></mtr></mtable></mrow></math>

Graph

It is convenient to define the Lagrange function, since the multipliers are unique under CC-LICQ, cf. [[6]],

Graph

We also use the corresponding tangent space

Graph

We now focus on the definition of nondegeneracy for M-stationary points, which was introduced in [[10]]. It is justified there by showing that all M-stationary points of CCOP are generically nondegenerate.

Definition 3

(Nondegenerate M-stationarity, see [[10]]) An M-stationary point <math xmlns="http://www.w3.org/1998/Math/MathML"><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></math> of CCOP is called nondegenerate if

NDM1: CC-LICQ holds at <math xmlns="http://www.w3.org/1998/Math/MathML"><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></math> ,

NDM2: <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mover accent="true"><mrow><mi>μ</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mi>q</mi></msub><mo>></mo><mn>0</mn></mrow></math> for all <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>q</mi><mo>∈</mo><msub><mi>Q</mi><mn>0</mn></msub><mrow><mo stretchy="false">(</mo><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo stretchy="false">)</mo></mrow></mrow></math> ,

NDM3: if <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mfenced close="∥" open="∥"><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mfenced><mn>0</mn></msub><mo><</mo><mi>s</mi></mrow></math> then <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mover accent="true"><mrow><mi>γ</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mi>i</mi></msub><mo>≠</mo><mn>0</mn></mrow></math> for all <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>i</mi><mo>∈</mo><msub><mi>I</mi><mn>0</mn></msub><mrow><mo stretchy="false">(</mo><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo stretchy="false">)</mo></mrow></mrow></math> ,

NDM4: the matrix <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mi>D</mi><mn>2</mn></msup><mi>L</mi><mrow><mo stretchy="false">(</mo><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo stretchy="false">)</mo></mrow><msub><mo>↾</mo><msub><mi mathvariant="script">T</mi><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></msub></msub></mrow></math> is nonsingular.

For a nondegenerate M-stationary point we eventually use an additional condition:

NDM5: <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>γ</mi><mi>i</mi></msub><mo>≠</mo><mn>0</mn></mrow></math> holds for all <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>i</mi><mo>∈</mo><msub><mi>I</mi><mn>0</mn></msub><mrow><mo stretchy="false">(</mo><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo stretchy="false">)</mo></mrow></mrow></math> .

With a nondegenerate M-stationary point <math xmlns="http://www.w3.org/1998/Math/MathML"><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></math> an M-index can be associated. The M-index captures the structure of CCOP locally around <math xmlns="http://www.w3.org/1998/Math/MathML"><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></math> and defines the type of an M-stationary point, see [[10]] for details. In particular, nondegenerate minimizers of CCOP are characterized by a vanishing M-index. If the M-index does not vanish, we get all kinds of saddle points.

Definition 4

(M-index, see [[10]]) Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></math> be a nondegenerate M-stationary point of CCOP. The number of negative eigenvalues of the matrix <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mi>D</mi><mn>2</mn></msup><mspace width="0.166667em" /><mi>L</mi><mrow><mo stretchy="false">(</mo><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo stretchy="false">)</mo></mrow><msub><mo>↾</mo><msub><mi mathvariant="script">T</mi><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></msub></msub></mrow></math> is called its quadratic index (QI). The number <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>s</mi><mo>-</mo><msub><mfenced close="∥" open="∥"><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mfenced><mn>0</mn></msub></mrow></math> is called the sparsity index (SI) of <math xmlns="http://www.w3.org/1998/Math/MathML"><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></math> . We define the M-index (MI) as the sum of both, i. e. <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>M</mi><mi>I</mi><mo>=</mo><mi>S</mi><mi>I</mi><mo>+</mo><mi>Q</mi><mi>I</mi></mrow></math> .

Now, we are ready to associate with CCOP the regularized continuous reformulation as suggested in [[9]]:

Graph

where the components of <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>c</mi><mo>∈</mo><msup><mrow><mi mathvariant="double-struck">R</mi></mrow><mi>n</mi></msup></mrow></math> are positive and pairwise different, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mn>0</mn><mo><</mo><mi>ε</mi><mo>≤</mo><mfrac><mn>1</mn><mrow><mi>n</mi><mo>-</mo><mi>s</mi></mrow></mfrac></mrow></math> . Given a feasible point <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo stretchy="false">)</mo></mrow></math> of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="script">R</mi></math> , we define the index sets which correspond to the orthogonality type constraints <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>x</mi><mi>i</mi></msub><msub><mi>y</mi><mi>i</mi></msub><mo>=</mo><mn>0</mn></mrow></math> , <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>y</mi><mi>i</mi></msub><mo>≥</mo><mn>0</mn></mrow></math> :

Graph

The index sets of the active inequality constraints will be denoted by

Graph

The regularized continuous reformulation <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="script">R</mi></math> is a special case of MPOC. The latter class was examined in [[4]], where the MPOC-tailored linear independence constraint qualification and the notion of (nondegenerate) T-stationary points with the corresponding T-index were introduced. It has been shown there that T-stationarity is the topologically relevant stationarity notion for MPOC, again in the sense of Morse theory. We note that the alternative concept of S-stationarity has been defined for the original continuous reformulation (1). It has been shown in [[4]] that S-stationarity implies T-stationarity for (1), but not vice versa. These both facts motivated us in [[9]] to apply T-, rather than S-stationarity to the regularization <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="script">R</mi></math> .

Definition 5

(MPOC-LICQ, [[9]]) We say that a feasible point <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo stretchy="false">)</mo></mrow></math> of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="script">R</mi></math> satisfies the MPOC-tailored linear independence constraint qualification (MPOC-LICQ) if the following vectors are linearly independent:

Graph

Definition 6

(T-stationary point, [[9]]) A feasible point <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo stretchy="false">)</mo></mrow></math> of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="script">R</mi></math> is called T-stationary if there exist multipliers

Graph

such that the following conditions hold:

5 <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtable><mtr><mtd columnalign="right"><mrow><mfenced close=")" open="("><mrow><mtable><mtr><mtd><mrow><mi mathvariant="normal">∇</mi><mi>f</mi><mo stretchy="false">(</mo><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo stretchy="false">)</mo></mrow></mtd></mtr><mtr><mtd><mrow><mrow /><mi>c</mi></mrow></mtd></mtr></mtable></mrow></mfenced><mo>=</mo></mrow></mtd><mtd columnalign="left"><mstyle displaystyle="true" scriptlevel="0"><mrow><mrow /><munder><mo movablelimits="false">∑</mo><mrow><mi>p</mi><mo>∈</mo><mi>P</mi></mrow></munder><msub><mover accent="true"><mrow><mi>λ</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mi>p</mi></msub><mfenced close=")" open="("><mrow><mtable><mtr><mtd><mrow><mi mathvariant="normal">∇</mi><msub><mi>h</mi><mi>p</mi></msub><mrow><mo stretchy="false">(</mo><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo stretchy="false">)</mo></mrow></mrow></mtd></mtr><mtr><mtd><mrow><mrow /><mn>0</mn></mrow></mtd></mtr></mtable></mrow></mfenced><mo>+</mo><munder><mo movablelimits="false">∑</mo><mrow><mi>q</mi><mo>∈</mo><msub><mi>Q</mi><mn>0</mn></msub><mrow><mo stretchy="false">(</mo><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo stretchy="false">)</mo></mrow></mrow></munder><msub><mover accent="true"><mrow><mi>μ</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mrow><mn>1</mn><mo>,</mo><mi>q</mi></mrow></msub><mfenced close=")" open="("><mrow><mtable><mtr><mtd><mrow><mi mathvariant="normal">∇</mi><msub><mi>g</mi><mi>q</mi></msub><mrow><mo stretchy="false">(</mo><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo stretchy="false">)</mo></mrow></mrow></mtd></mtr><mtr><mtd><mrow><mrow /><mn>0</mn></mrow></mtd></mtr></mtable></mrow></mfenced><mo>-</mo><munder><mo movablelimits="false">∑</mo><mrow><mi>i</mi><mo>∈</mo><mi mathvariant="script">E</mi><mo stretchy="false">(</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo stretchy="false">)</mo></mrow></munder><msub><mover accent="true"><mrow><mi>μ</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mrow><mn>2</mn><mo>,</mo><mi>i</mi></mrow></msub><mfenced close=")" open="("><mrow><mtable><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mrow><mrow /><msub><mi>e</mi><mi>i</mi></msub></mrow></mtd></mtr></mtable></mrow></mfenced></mrow></mstyle></mtd></mtr><mtr><mtd columnalign="right"><mrow /></mtd><mtd columnalign="left"><mstyle displaystyle="true" scriptlevel="0"><mrow><mrow /><mo>+</mo><msub><mover accent="true"><mrow><mi>μ</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mn>3</mn></msub><mfenced close=")" open="("><mrow><mtable><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mrow><mrow /><mi>e</mi></mrow></mtd></mtr></mtable></mrow></mfenced><mo>+</mo><munder><mo movablelimits="false">∑</mo><mrow><mi>i</mi><mo>∈</mo><msub><mi>a</mi><mn>01</mn></msub><mfenced close=")" open="("><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mfenced></mrow></munder><msub><mover accent="true"><mrow><mi>σ</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mrow><mn>1</mn><mo>,</mo><mi>i</mi></mrow></msub><mfenced close=")" open="("><mrow><mtable><mtr><mtd><msub><mi>e</mi><mi>i</mi></msub></mtd></mtr><mtr><mtd><mrow><mrow /><mn>0</mn></mrow></mtd></mtr></mtable></mrow></mfenced><mo>+</mo><munder><mo movablelimits="false">∑</mo><mrow><mi>i</mi><mo>∈</mo><msub><mi>a</mi><mn>10</mn></msub><mfenced close=")" open="("><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mfenced></mrow></munder><msub><mover accent="true"><mrow><mi>σ</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mrow><mn>2</mn><mo>,</mo><mi>i</mi></mrow></msub><mfenced close=")" open="("><mrow><mtable><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mrow><mrow /><msub><mi>e</mi><mi>i</mi></msub></mrow></mtd></mtr></mtable></mrow></mfenced></mrow></mstyle></mtd></mtr><mtr><mtd columnalign="right"><mrow /></mtd><mtd columnalign="left"><mstyle displaystyle="true" scriptlevel="0"><mrow><mrow /><mo>+</mo><munder><mo movablelimits="false">∑</mo><mrow><mi>i</mi><mo>∈</mo><msub><mi>a</mi><mn>00</mn></msub><mfenced close=")" open="("><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mfenced></mrow></munder><mfenced close=")" open="("><msub><mover accent="true"><mrow><mi>ϱ</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mrow><mn>1</mn><mo>,</mo><mi>i</mi></mrow></msub><mfenced close=")" open="("><mrow><mtable><mtr><mtd><msub><mi>e</mi><mi>i</mi></msub></mtd></mtr><mtr><mtd><mrow><mrow /><mn>0</mn></mrow></mtd></mtr></mtable></mrow></mfenced><mo>+</mo><msub><mover accent="true"><mrow><mi>ϱ</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mrow><mn>2</mn><mo>,</mo><mi>i</mi></mrow></msub><mfenced close=")" open="("><mrow><mtable><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mrow><mrow /><msub><mi>e</mi><mi>i</mi></msub></mrow></mtd></mtr></mtable></mrow></mfenced></mfenced><mo>,</mo></mrow></mstyle></mtd></mtr></mtable></mrow></math>

Graph

6 <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtable><mtr><mtd columnalign="right"><mrow><msub><mover accent="true"><mrow><mi>μ</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mrow><mn>1</mn><mo>,</mo><mi>q</mi></mrow></msub><mo>≥</mo></mrow></mtd><mtd columnalign="left"><mrow><mrow /><mn>0</mn><mo>,</mo><mspace width="0.166667em" /><mi>q</mi><mo>∈</mo><msub><mi>Q</mi><mn>0</mn></msub><mfenced close=")" open="("><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mfenced><mo>,</mo><mspace width="1em" /><msub><mover accent="true"><mrow><mi>μ</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mrow><mn>2</mn><mo>,</mo><mi>i</mi></mrow></msub><mo>≥</mo><mn>0</mn><mo>,</mo><mspace width="0.166667em" /><mi>i</mi><mo>∈</mo><mi mathvariant="script">E</mi><mrow><mo stretchy="false">(</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo stretchy="false">)</mo></mrow><mo>,</mo><mspace width="1em" /></mrow></mtd></mtr><mtr><mtd columnalign="right"><mrow><mrow /><msub><mover accent="true"><mrow><mi>μ</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mn>3</mn></msub><mo>≥</mo></mrow></mtd><mtd columnalign="left"><mrow><mrow /><mn>0</mn><mo>,</mo><mspace width="0.166667em" /><msub><mover accent="true"><mrow><mi>μ</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mn>3</mn></msub><mfenced close=")" open="("><munderover><mo movablelimits="false">∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msub><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mi>i</mi></msub><mo>-</mo><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>-</mo><mi>s</mi><mo stretchy="false">)</mo></mrow></mfenced><mo>=</mo><mn>0</mn><mo>,</mo></mrow></mtd></mtr></mtable></mrow></math>

Graph

7 <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtable><mtr><mtd columnalign="right"><mrow><msub><mover accent="true"><mrow><mi>ϱ</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mrow><mn>1</mn><mo>,</mo><mi>i</mi></mrow></msub><mo>=</mo></mrow></mtd><mtd columnalign="left"><mrow><mrow /><mn>0</mn><mspace width="0.333333em" /><mtext>or</mtext><mspace width="0.333333em" /><msub><mover accent="true"><mrow><mi>ϱ</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mrow><mn>2</mn><mo>,</mo><mi>i</mi></mrow></msub><mo>≤</mo><mn>0</mn><mo>,</mo><mi>i</mi><mo>∈</mo><msub><mi>a</mi><mn>00</mn></msub><mfenced close=")" open="("><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mfenced><mo>.</mo></mrow></mtd></mtr></mtable></mrow></math>

Graph

We define the appropriate Lagrange function:

Graph

Moreover, we set for the corresponding tangent space

Graph

Definition 7

(Nondegenerate T-stationary point, [[9]]) A T-stationary point <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo stretchy="false">)</mo></mrow></math> of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="script">R</mi></math> with multipliers <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><mover accent="true"><mrow><mi>λ</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>μ</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>σ</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>ϱ</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo stretchy="false">)</mo></mrow></math> is called nondegenerate if

NDT1: MPOC-LICQ holds at <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo stretchy="false">)</mo></mrow></math> ,

NDT2: <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mover accent="true"><mrow><mi>μ</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mrow><mn>1</mn><mo>,</mo><mi>q</mi></mrow></msub><mo>></mo><mn>0</mn></mrow></math> , <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>q</mi><mo>∈</mo><msub><mi>Q</mi><mn>0</mn></msub><mfenced close=")" open="("><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mfenced></mrow></math> , <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mover accent="true"><mrow><mi>μ</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mrow><mn>2</mn><mo>,</mo><mi>i</mi></mrow></msub><mo>></mo><mn>0</mn></mrow></math> , <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>i</mi><mo>∈</mo><mi mathvariant="script">E</mi><mfenced close=")" open="("><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mfenced></mrow></math> , and <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mover accent="true"><mrow><mi>μ</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mn>3</mn></msub><mo>></mo><mn>0</mn></mrow></math> if <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></msubsup><msub><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mi>i</mi></msub><mo>=</mo><mi>n</mi><mo>-</mo><mi>s</mi></mrow></math> ,

NDT3: <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mover accent="true"><mrow><mi>ϱ</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mrow><mn>1</mn><mo>,</mo><mi>i</mi></mrow></msub><mo>≠</mo><mn>0</mn></mrow></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mover accent="true"><mrow><mi>ϱ</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mrow><mn>2</mn><mo>,</mo><mi>i</mi></mrow></msub><mo><</mo><mn>0</mn></mrow></math> , <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>i</mi><mo>∈</mo><msub><mi>a</mi><mn>00</mn></msub><mfenced close=")" open="("><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mfenced></mrow></math> ,

NDT4: the matrix <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mi>D</mi><mn>2</mn></msup><mspace width="0.166667em" /><msup><mi>L</mi><mi mathvariant="script">R</mi></msup><mrow><mo stretchy="false">(</mo><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo stretchy="false">)</mo></mrow><msub><mo>↾</mo><msubsup><mrow><mi mathvariant="script">T</mi></mrow><mrow><mo stretchy="false">(</mo><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo stretchy="false">)</mo></mrow><mi mathvariant="script">R</mi></msubsup></msub></mrow></math> is nonsingular.

For a nondegenerate T-stationary point we eventually use additional conditions:

NDT5: if <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>a</mi><mn>00</mn></msub><mfenced close=")" open="("><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mfenced><mo>≠</mo><mi mathvariant="normal">∅</mi></mrow></math> , then <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mover accent="true"><mrow><mi>σ</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mrow><mn>1</mn><mo>,</mo><mi>i</mi></mrow></msub><mo>≠</mo><mn>0</mn></mrow></math> for all <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>i</mi><mo>∈</mo><msub><mi>a</mi><mn>01</mn></msub><mrow><mo stretchy="false">(</mo><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo stretchy="false">)</mo></mrow></mrow></math> .

NDT6: <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mover accent="true"><mrow><mi>σ</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mrow><mn>1</mn><mo>,</mo><mi>i</mi></mrow></msub><mo>≠</mo><mn>0</mn></mrow></math> , <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>i</mi><mo>∈</mo><msub><mi>a</mi><mn>01</mn></msub><mrow><mo stretchy="false">(</mo><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo stretchy="false">)</mo></mrow></mrow></math> .

Definition 8

(T-index, [[9]]) Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo stretchy="false">)</mo></mrow></math> be a nondegenerate T-stationary point of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="script">R</mi></math> with unique multipliers <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close=")" open="("><mover accent="true"><mrow><mi>λ</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>μ</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>σ</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>ϱ</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mfenced></math> . The number of negative eigenvalues of the matrix <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mi>D</mi><mn>2</mn></msup><mspace width="0.166667em" /><msup><mi>L</mi><mi mathvariant="script">R</mi></msup><mrow><mo stretchy="false">(</mo><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo stretchy="false">)</mo></mrow><msub><mo>↾</mo><msubsup><mrow><mi mathvariant="script">T</mi></mrow><mrow><mo stretchy="false">(</mo><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo stretchy="false">)</mo></mrow><mi mathvariant="script">R</mi></msubsup></msub></mrow></math> is called its quadratic index (QI). The cardinality of <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>a</mi><mn>00</mn></msub><mfenced close=")" open="("><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mfenced></mrow></math> is called the biactive index (BI) of <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo stretchy="false">)</mo></mrow></math> . We define the T-index (TI) as the sum of both, i.e. <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>T</mi><mi>I</mi><mo>=</mo><mi>Q</mi><mi>I</mi><mo>+</mo><mi>B</mi><mi>I</mi></mrow></math> .

The nondegeneracy conditions NDT1-NDT4 are tailored for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="script">R</mi></math> . Note that NDT2 corresponds to the strict complementarity and NDT4 to the second-order regularity as they are typically defined in the context of nonlinear programming. NDT1 substitutes the usual linear independence constraint qualification. NDT3 is new and says that the multipliers corresponding to biactive orthogonality type constraints must not vanish. With a nondegenerate T-stationary point <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo stretchy="false">)</mo></mrow></math> a T-index can be associated. The T-index captures the structure of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="script">R</mi></math> locally around <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo stretchy="false">)</mo></mrow></math> and defines the type of a T-stationary point, see [[9]] for details. In particular, nondegenerate minimizers of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="script">R</mi></math> are characterized by a vanishing T-index. If the T-index does not vanish, we get all kinds of saddle points.

Next Lemma 1 provides insights into the structure of auxiliary y-variables corresponding to a T-stationary point of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="script">R</mi></math> .

Lemma 1

(Auxiliary y-variables in <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="script">R</mi></math> , [[9]]) Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo stretchy="false">)</mo></mrow></math> be a T-stationary point of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="script">R</mi></math> , then it holds:

the summation inequality constraint is active, i.e. <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></msubsup><msub><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mi>i</mi></msub><mo>=</mo><mi>n</mi><mo>-</mo><mi>s</mi></mrow></math> ,

the index set <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>a</mi><mn>01</mn></msub><mrow><mo stretchy="false">(</mo><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo stretchy="false">)</mo></mrow></mrow></math> consists of exactly <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>n</mi><mo>-</mo><mi>s</mi></mrow></math> elements,

<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>n</mi><mo>-</mo><mi>s</mi><mo>-</mo><mn>1</mn></mrow></math> components of <math xmlns="http://www.w3.org/1998/Math/MathML"><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></math> are equal to <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mn>1</mn><mo>+</mo><mi>ε</mi></mrow></math> , one component is equal to <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mn>1</mn><mo>-</mo><mo stretchy="false">(</mo><mi>n</mi><mo>-</mo><mi>s</mi><mo>-</mo><mn>1</mn><mo stretchy="false">)</mo><mi>ε</mi></mrow></math> , and s remaining components vanish.

We note that nondegenerate M-stationary points of CCOP naturally correspond to nondegenerate T-stationary points of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="script">R</mi></math> and vice versa. As shown in [[9]], also their M- and T-indices coincide. Thus, the regularized continuous reformulation <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="script">R</mi></math> can be likewise studied instead of (1).

Theorem 1

(Stationarity of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="script">R</mi></math> and CCOP, [[9]])

If <math xmlns="http://www.w3.org/1998/Math/MathML"><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></math> is an M-stationary point of CCOP, then there exist at least <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close=")" open="("><mrow><mtable><mtr><mtd><mrow><mi>n</mi><mo>-</mo><msub><mfenced close="∥" open="∥"><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mfenced><mn>0</mn></msub><mo>-</mo><mn>1</mn></mrow></mtd></mtr><mtr><mtd><mrow><mrow /><mi>n</mi><mo>-</mo><mi>s</mi><mo>-</mo><mn>1</mn></mrow></mtd></mtr></mtable></mrow></mfenced></math> choices of <math xmlns="http://www.w3.org/1998/Math/MathML"><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo stretchy="false">)</mo></mrow></math> is a T-stationary point of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="script">R</mi></math> . If <math xmlns="http://www.w3.org/1998/Math/MathML"><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></math> is additionally nondegenerate with M-index m, then all corresponding T-stationary points <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo stretchy="false">)</mo></mrow></math> are also nondegenerate with T-index m. Moreover, their number is exactly <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close=")" open="("><mrow><mtable><mtr><mtd><mrow><mi>n</mi><mo>-</mo><msub><mfenced close="∥" open="∥"><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mfenced><mn>0</mn></msub><mo>-</mo><mn>1</mn></mrow></mtd></mtr><mtr><mtd><mrow><mrow /><mi>n</mi><mo>-</mo><mi>s</mi><mo>-</mo><mn>1</mn></mrow></mtd></mtr></mtable></mrow></mfenced></math> , and NDT5 holds at any of them.

If <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo stretchy="false">)</mo></mrow></math> is a T-stationary point of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="script">R</mi></math> , then <math xmlns="http://www.w3.org/1998/Math/MathML"><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></math> is an M-stationary point of CCOP. If <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo stretchy="false">)</mo></mrow></math> is additionally nondegenerate with T-index m and satisfies NDT5, then <math xmlns="http://www.w3.org/1998/Math/MathML"><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></math> is also nondegenerate with M-index m.

Scholtes-type regularization

Let us now regularize the orthogonality type constraints in <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="script">R</mi></math> by using the Scholtes' idea, cf. [[1]]:

Graph

where <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>t</mi><mo>></mo><mn>0</mn></mrow></math> . Note that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="script">S</mi></math> from above falls into the scope of nonlinear programming. The notation for the sets <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>Q</mi><mn>0</mn></msub><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="script">E</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></math> , which were used for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="script">R</mi></math> , will be used here again. Furthermore, we define for a feasible point <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close=")" open="("><mi>x</mi><mo>,</mo><mi>y</mi></mfenced></math> of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="script">S</mi></math> the index set of vanishing y-components as well as the index sets of active relaxed orthogonality type constraints:

Graph

We also eventually use the following index sets:

Graph

For the sake of completeness we state the linear independence constraint qualification for the nonlinear programming problem <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="script">S</mi></math> .

Definition 9

(LICQ) We say that a feasible point (x, y) of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="script">S</mi></math> satisfies the linear independence constraint qualification (LICQ) if the following vectors are linearly independent:

Graph

Let us relate MPOC-LICQ for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="script">R</mi></math> with LICQ for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="script">S</mi></math> .

Theorem 2

(MPOC-LICQ vs. LICQ) Let a feasible point <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo stretchy="false">)</mo></mrow></math> of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="script">R</mi></math> fulfill MPOC-LICQ. Then, LICQ holds at all feasible points (x, y) of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="script">S</mi></math> for all sufficiently small t, whenever they are sufficiently close to <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo stretchy="false">)</mo></mrow></math> .

Proof

Let us contrarily assume that there exists a sequence of feasible points <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close=")" open="("><msup><mi>x</mi><mi>t</mi></msup><mo>,</mo><msup><mi>y</mi><mi>t</mi></msup></mfenced></math> of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="script">S</mi></math> violating LICQ, which converges to <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo stretchy="false">)</mo></mrow></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>t</mi><mo stretchy="false">→</mo><mn>0</mn></mrow></math> . Additionally, suppose that along some subsequence, which we index by t again, it holds <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></msubsup><msubsup><mi>y</mi><mi>i</mi><mi>t</mi></msubsup><mo>=</mo><mi>n</mi><mo>-</mo><mi>s</mi></mrow></math> . Then, we have <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></msubsup><msub><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mi>i</mi></msub><mo>=</mo><mi>n</mi><mo>-</mo><mi>s</mi></mrow></math> . Due to MPOC-LICQ at <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close=")" open="("><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mfenced></math> as well as continuity of <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="normal">∇</mi><mi>h</mi></mrow></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="normal">∇</mi><mi>g</mi></mrow></math> , we have that for t sufficiently small all multipliers <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mrow><mover accent="true"><mrow><mi>λ</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mrow><mi>t</mi></msup><mo>,</mo><msup><mrow><mover accent="true"><mrow><mi>μ</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mrow><mi>t</mi></msup><mo>,</mo><msup><mrow><mover accent="true"><mrow><mi>σ</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mrow><mi>t</mi></msup><mo>,</mo><msup><mrow><mover accent="true"><mrow><mi>ϱ</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mrow><mi>t</mi></msup></mrow></math> in the following equation vanish:

8 <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtable><mtr><mtd columnalign="right"><mrow><mfenced close=")" open="("><mrow><mtable><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mrow><mrow /><mn>0</mn></mrow></mtd></mtr></mtable></mrow></mfenced><mo>=</mo></mrow></mtd><mtd columnalign="left"><mstyle displaystyle="true" scriptlevel="0"><mrow><mrow /><munder><mo movablelimits="false">∑</mo><mrow><mi>p</mi><mo>∈</mo><mi>P</mi></mrow></munder><msubsup><mrow><mover accent="true"><mrow><mi>λ</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mrow><mi>p</mi><mi>t</mi></msubsup><mfenced close=")" open="("><mrow><mtable><mtr><mtd><mrow><mi mathvariant="normal">∇</mi><msub><mi>h</mi><mi>p</mi></msub><mfenced close=")" open="("><msup><mi>x</mi><mi>t</mi></msup></mfenced></mrow></mtd></mtr><mtr><mtd><mrow><mrow /><mn>0</mn></mrow></mtd></mtr></mtable></mrow></mfenced><mo>+</mo><munder><mo movablelimits="false">∑</mo><mrow><mi>q</mi><mo>∈</mo><msub><mi>Q</mi><mn>0</mn></msub><mrow><mo stretchy="false">(</mo><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo stretchy="false">)</mo></mrow></mrow></munder><msubsup><mrow><mover accent="true"><mrow><mi>μ</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mrow><mrow><mn>1</mn><mo>,</mo><mi>q</mi></mrow><mi>t</mi></msubsup><mfenced close=")" open="("><mrow><mtable><mtr><mtd><mrow><mi mathvariant="normal">∇</mi><msub><mi>g</mi><mi>q</mi></msub><mfenced close=")" open="("><msup><mi>x</mi><mi>t</mi></msup></mfenced></mrow></mtd></mtr><mtr><mtd><mrow><mrow /><mn>0</mn></mrow></mtd></mtr></mtable></mrow></mfenced><mo>-</mo><munder><mo movablelimits="false">∑</mo><mrow><mi>i</mi><mo>∈</mo><mi mathvariant="script">E</mi><mo stretchy="false">(</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo stretchy="false">)</mo></mrow></munder><msubsup><mrow><mover accent="true"><mrow><mi>μ</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mrow><mrow><mn>2</mn><mo>,</mo><mi>i</mi></mrow><mi>t</mi></msubsup><mfenced close=")" open="("><mrow><mtable><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mrow><mrow /><msub><mi>e</mi><mi>i</mi></msub></mrow></mtd></mtr></mtable></mrow></mfenced></mrow></mstyle></mtd></mtr><mtr><mtd columnalign="right"><mrow /></mtd><mtd columnalign="left"><mstyle displaystyle="true" scriptlevel="0"><mrow><mrow /><mo>+</mo><msubsup><mrow><mover accent="true"><mrow><mi>μ</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mrow><mn>3</mn><mi>t</mi></msubsup><mfenced close=")" open="("><mrow><mtable><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mrow><mrow /><mi>e</mi></mrow></mtd></mtr></mtable></mrow></mfenced><mo>+</mo><munder><mo movablelimits="false">∑</mo><mrow><mi>i</mi><mo>∈</mo><msub><mi>a</mi><mn>01</mn></msub><mfenced close=")" open="("><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mfenced></mrow></munder><msubsup><mrow><mover accent="true"><mrow><mi>σ</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mrow><mrow><mn>1</mn><mo>,</mo><mi>i</mi></mrow><mi>t</mi></msubsup><mfenced close=")" open="("><mrow><mtable><mtr><mtd><mrow><msubsup><mi>y</mi><mrow><mi>i</mi></mrow><mi>t</mi></msubsup><msub><mi>e</mi><mi>i</mi></msub></mrow></mtd></mtr><mtr><mtd><mrow><mrow /><msubsup><mi>x</mi><mrow><mi>i</mi></mrow><mi>t</mi></msubsup><msub><mi>e</mi><mi>i</mi></msub></mrow></mtd></mtr></mtable></mrow></mfenced><mo>+</mo><munder><mo movablelimits="false">∑</mo><mrow><mi>i</mi><mo>∈</mo><msub><mi>a</mi><mn>10</mn></msub><mfenced close=")" open="("><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mfenced></mrow></munder><msubsup><mrow><mover accent="true"><mrow><mi>σ</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mrow><mrow><mn>2</mn><mo>,</mo><mi>i</mi></mrow><mi>t</mi></msubsup><mfenced close=")" open="("><mrow><mtable><mtr><mtd><mrow><msubsup><mi>y</mi><mrow><mi>i</mi></mrow><mi>t</mi></msubsup><msub><mi>e</mi><mi>i</mi></msub></mrow></mtd></mtr><mtr><mtd><mrow><mrow /><msubsup><mi>x</mi><mrow><mi>i</mi></mrow><mi>t</mi></msubsup><msub><mi>e</mi><mi>i</mi></msub></mrow></mtd></mtr></mtable></mrow></mfenced></mrow></mstyle></mtd></mtr><mtr><mtd columnalign="right"><mrow /></mtd><mtd columnalign="left"><mstyle displaystyle="true" scriptlevel="0"><mrow><mrow /><mo>+</mo><munder><mo movablelimits="false">∑</mo><mrow><mi>i</mi><mo>∈</mo><msub><mi>a</mi><mn>00</mn></msub><mfenced close=")" open="("><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mfenced></mrow></munder><mfenced close=")" open="("><msubsup><mrow><mover accent="true"><mrow><mi>ϱ</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mrow><mrow><mn>1</mn><mo>,</mo><mi>i</mi></mrow><mi>t</mi></msubsup><mfenced close=")" open="("><mrow><mtable><mtr><mtd><msub><mi>e</mi><mi>i</mi></msub></mtd></mtr><mtr><mtd><mrow><mrow /><mn>0</mn></mrow></mtd></mtr></mtable></mrow></mfenced><mo>+</mo><msubsup><mrow><mover accent="true"><mrow><mi>ϱ</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mrow><mrow><mn>2</mn><mo>,</mo><mi>i</mi></mrow><mi>t</mi></msubsup><mfenced close=")" open="("><mrow><mtable><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mrow><mrow /><msub><mi>e</mi><mi>i</mi></msub></mrow></mtd></mtr></mtable></mrow></mfenced></mfenced><mo>.</mo></mrow></mstyle></mtd></mtr></mtable></mrow></math>

Graph

Moreover, due to the violation of LICQ at <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close=")" open="("><msup><mi>x</mi><mi>t</mi></msup><mo>,</mo><msup><mi>y</mi><mi>t</mi></msup></mfenced></math> , there exist multipliers <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mi>λ</mi><mi>t</mi></msup><mo>,</mo><msup><mi>μ</mi><mi>t</mi></msup><mo>,</mo><msup><mi>η</mi><mi>t</mi></msup><mo>,</mo><msup><mi>ν</mi><mi>t</mi></msup></mrow></math> , not all vanishing, with

Graph

For t sufficiently small we have <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>Q</mi><mn>0</mn></msub><mfenced close=")" open="("><msup><mi>x</mi><mi>t</mi></msup></mfenced><mo>⊂</mo><msub><mi>Q</mi><mn>0</mn></msub><mrow><mo stretchy="false">(</mo><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo stretchy="false">)</mo></mrow></mrow></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="script">E</mi><mfenced close=")" open="("><msup><mi>y</mi><mi>t</mi></msup></mfenced><mo>⊂</mo><mi mathvariant="script">E</mi><mfenced close=")" open="("><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mfenced></mrow></math> . In addition, it holds <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="script">H</mi><mfenced close=")" open="("><msup><mi>x</mi><mi>t</mi></msup><mo>,</mo><msup><mi>y</mi><mi>t</mi></msup></mfenced><mo>⊂</mo><msub><mi>a</mi><mn>01</mn></msub><mrow><mo stretchy="false">(</mo><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo stretchy="false">)</mo></mrow><mo>∪</mo><msub><mi>a</mi><mn>10</mn></msub><mrow><mo stretchy="false">(</mo><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo stretchy="false">)</mo></mrow><mo>∪</mo><msub><mi>a</mi><mn>00</mn></msub><mrow><mo stretchy="false">(</mo><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo stretchy="false">)</mo></mrow></mrow></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="script">N</mi><mfenced close=")" open="("><msup><mi>y</mi><mi>t</mi></msup></mfenced><mo>⊂</mo><msub><mi>a</mi><mn>10</mn></msub><mrow><mo stretchy="false">(</mo><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo stretchy="false">)</mo></mrow><mo>∪</mo><msub><mi>a</mi><mn>00</mn></msub><mrow><mo stretchy="false">(</mo><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo stretchy="false">)</mo></mrow></mrow></math> . By setting some <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>μ</mi></math> -multipliers to be zero if needed, we equivalently obtain:

Graph

This, however, implies that not all multipliers in the following equation vanish:

Graph

A contradiction to (8) follows by taking into account that <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="script">H</mi><mrow><mo stretchy="false">(</mo><msup><mi>x</mi><mi>t</mi></msup><mo>,</mo><msup><mi>y</mi><mi>t</mi></msup><mo stretchy="false">)</mo></mrow><mo>∩</mo><mi mathvariant="script">N</mi><mrow><mo stretchy="false">(</mo><msup><mi>y</mi><mi>t</mi></msup><mo stretchy="false">)</mo></mrow><mo>=</mo><mi mathvariant="normal">∅</mi></mrow></math> . If instead we suppose that there is no subsequence with <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><munderover><mo movablelimits="false">∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msubsup><mi>y</mi><mi>i</mi><mi>t</mi></msubsup><mo>=</mo><mi>n</mi><mo>-</mo><mi>s</mi></mrow></math> , then we can consider a subsequence with <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></msubsup><msubsup><mi>y</mi><mi>i</mi><mi>t</mi></msubsup><mo>></mo><mi>n</mi><mo>-</mo><mi>s</mi></mrow></math> . By following a similar argumentation, we produce a contradiction to (8) again. <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>□</mo></math>

Next, we give the definitions of a (nondegenerate) Karush–Kuhn–Tucker point of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="script">S</mi></math> and of its quadratic index as it is meanwhile standard in nonlinear programming, see e.g. [[8]].

Definition 10

(Karush–Kuhn–Tucker point) A feasible point (x, y) of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="script">S</mi></math> is called Kurush–Kuhn–Tucker point if there exist multipliers

Graph

such that the following conditions hold:

9 <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtable><mtr><mtd columnalign="right"><mrow><mfenced close=")" open="("><mrow><mtable><mtr><mtd><mrow><mi mathvariant="normal">∇</mi><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mtd></mtr><mtr><mtd><mrow><mrow /><mi>c</mi></mrow></mtd></mtr></mtable></mrow></mfenced><mo>=</mo></mrow></mtd><mtd columnalign="left"><mstyle displaystyle="true" scriptlevel="0"><mrow><mrow /><munder><mo movablelimits="false">∑</mo><mrow><mi>p</mi><mo>∈</mo><mi>P</mi></mrow></munder><msub><mi>λ</mi><mi>p</mi></msub><mfenced close=")" open="("><mrow><mtable><mtr><mtd><mrow><mi mathvariant="normal">∇</mi><msub><mi>h</mi><mi>p</mi></msub><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></mtd></mtr><mtr><mtd><mrow><mrow /><mn>0</mn></mrow></mtd></mtr></mtable></mrow></mfenced><mo>+</mo><munder><mo movablelimits="false">∑</mo><mrow><mi>q</mi><mo>∈</mo><msub><mi>Q</mi><mn>0</mn></msub><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></munder><msub><mi>μ</mi><mrow><mn>1</mn><mo>,</mo><mi>q</mi></mrow></msub><mfenced close=")" open="("><mrow><mtable><mtr><mtd><mrow><mi mathvariant="normal">∇</mi><msub><mi>g</mi><mi>q</mi></msub><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></mtd></mtr><mtr><mtd><mrow><mrow /><mn>0</mn></mrow></mtd></mtr></mtable></mrow></mfenced></mrow></mstyle></mtd></mtr><mtr><mtd columnalign="right"><mrow /></mtd><mtd columnalign="left"><mrow><mrow /><mo>-</mo><munder><mo movablelimits="false">∑</mo><mrow><mi>i</mi><mo>∈</mo><mi mathvariant="script">E</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></munder><msub><mi>μ</mi><mrow><mn>2</mn><mo>,</mo><mi>i</mi></mrow></msub><mfenced close=")" open="("><mrow><mtable><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mrow><mrow /><msub><mi>e</mi><mi>i</mi></msub></mrow></mtd></mtr></mtable></mrow></mfenced><mo>+</mo><msub><mi>μ</mi><mn>3</mn></msub><mfenced close=")" open="("><mrow><mtable><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mrow><mrow /><mi>e</mi></mrow></mtd></mtr></mtable></mrow></mfenced></mrow></mtd></mtr><mtr><mtd columnalign="right"><mrow /></mtd><mtd columnalign="left"><mstyle displaystyle="true" scriptlevel="0"><mrow><mrow /><mo>+</mo><munder><mo movablelimits="false">∑</mo><mrow><mi>i</mi><mo>∈</mo><msup><mrow><mi mathvariant="script">H</mi></mrow><mo>≥</mo></msup><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow></munder><msubsup><mi>η</mi><mi>i</mi><mo>≥</mo></msubsup><mfenced close=")" open="("><mrow><mtable><mtr><mtd><mrow><msub><mi>y</mi><mi>i</mi></msub><msub><mi>e</mi><mi>i</mi></msub></mrow></mtd></mtr><mtr><mtd><mrow><mrow /><msub><mi>x</mi><mi>i</mi></msub><msub><mi>e</mi><mi>i</mi></msub></mrow></mtd></mtr></mtable></mrow></mfenced><mo>-</mo><munder><mo movablelimits="false">∑</mo><mrow><mi>i</mi><mo>∈</mo><msup><mrow><mi mathvariant="script">H</mi></mrow><mo>≤</mo></msup><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow></munder><msubsup><mi>η</mi><mi>i</mi><mo>≤</mo></msubsup><mfenced close=")" open="("><mrow><mtable><mtr><mtd><mrow><msub><mi>y</mi><mi>i</mi></msub><msub><mi>e</mi><mi>i</mi></msub></mrow></mtd></mtr><mtr><mtd><mrow><mrow /><msub><mi>x</mi><mi>i</mi></msub><msub><mi>e</mi><mi>i</mi></msub></mrow></mtd></mtr></mtable></mrow></mfenced><mo>+</mo><munder><mo movablelimits="false">∑</mo><mrow><mi>i</mi><mo>∈</mo><mi mathvariant="script">N</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></munder><msub><mi>ν</mi><mi>i</mi></msub><mfenced close=")" open="("><mrow><mtable><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mrow><mrow /><msub><mi>e</mi><mi>i</mi></msub></mrow></mtd></mtr></mtable></mrow></mfenced><mo>,</mo></mrow></mstyle></mtd></mtr></mtable></mrow></math>

Graph

10 <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtable><mtr><mtd columnalign="right"><mrow><msub><mi>μ</mi><mrow><mn>1</mn><mo>,</mo><mi>q</mi></mrow></msub><mo>≥</mo></mrow></mtd><mtd columnalign="left"><mrow><mrow /><mn>0</mn><mo>,</mo><mspace width="0.166667em" /><mi>q</mi><mo>∈</mo><msub><mi>Q</mi><mn>0</mn></msub><mfenced close=")" open="("><mi>x</mi></mfenced><mo>,</mo><mspace width="1em" /><msub><mi>μ</mi><mrow><mn>2</mn><mo>,</mo><mi>i</mi></mrow></msub><mo>≥</mo><mn>0</mn><mo>,</mo><mspace width="0.166667em" /><mi>i</mi><mo>∈</mo><mi mathvariant="script">E</mi><mrow><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><mo>,</mo><mspace width="1em" /></mrow></mtd></mtr><mtr><mtd columnalign="right"><mrow><mrow /><msub><mi>μ</mi><mn>3</mn></msub><mo>≥</mo></mrow></mtd><mtd columnalign="left"><mrow><mrow /><mn>0</mn><mo>,</mo><msub><mi>μ</mi><mn>3</mn></msub><mfenced close=")" open="("><munderover><mo movablelimits="false">∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msub><mi>y</mi><mi>i</mi></msub><mo>-</mo><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>-</mo><mi>s</mi><mo stretchy="false">)</mo></mrow></mfenced><mo>=</mo><mn>0</mn><mo>,</mo></mrow></mtd></mtr></mtable></mrow></math>

Graph

11 <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtable><mtr><mtd columnalign="right"><mrow><msubsup><mi>η</mi><mi>i</mi><mo>≥</mo></msubsup><mo>≥</mo></mrow></mtd><mtd columnalign="left"><mrow><mrow /><mn>0</mn><mo>,</mo><mspace width="0.166667em" /><mi>i</mi><mo>∈</mo><msup><mrow><mi mathvariant="script">H</mi></mrow><mo>≥</mo></msup><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><mo>,</mo><mspace width="1em" /><msubsup><mi>η</mi><mi>i</mi><mo>≤</mo></msubsup><mo>≥</mo><mn>0</mn><mo>,</mo><mspace width="0.166667em" /><mi>i</mi><mo>∈</mo><msup><mrow><mi mathvariant="script">H</mi></mrow><mo>≤</mo></msup><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><mo>,</mo><mspace width="1em" /><msub><mi>ν</mi><mi>i</mi></msub><mo>≥</mo><mn>0</mn><mo>,</mo><mspace width="0.166667em" /><mi>i</mi><mo>∈</mo><mi mathvariant="script">N</mi><mrow><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><mo>.</mo></mrow></mtd></mtr><mtr><mtd columnalign="right"><mrow /></mtd></mtr></mtable></mrow></math>

Graph

We again define the Lagrange function as

Graph

The tangent space is given by

Graph

Definition 11

(Nondegenerate Karush–Kuhn–Tucker point) A Karush–Kuhn–Tucker point (x, y) of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="script">S</mi></math> with multipliers <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><mi>λ</mi><mo>,</mo><mi>μ</mi><mo>,</mo><mi>η</mi><mo>,</mo><mi>ν</mi><mo stretchy="false">)</mo></mrow></math> is called nondegenerate if

ND1: LICQ holds at (x, y),

ND2: <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>μ</mi><mrow><mn>1</mn><mo>,</mo><mi>q</mi></mrow></msub><mo>></mo><mn>0</mn></mrow></math> , <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>q</mi><mo>∈</mo><msub><mi>Q</mi><mn>0</mn></msub><mfenced close=")" open="("><mi>x</mi></mfenced></mrow></math> , <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>μ</mi><mrow><mn>2</mn><mo>,</mo><mi>i</mi></mrow></msub><mo>></mo><mn>0</mn></mrow></math> , <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>i</mi><mo>∈</mo><mi mathvariant="script">E</mi><mfenced close=")" open="("><mi>y</mi></mfenced></mrow></math> , <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mi>η</mi><mi>i</mi><mo>≥</mo></msubsup><mo>></mo><mn>0</mn></mrow></math> , <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>i</mi><mo>∈</mo><msup><mrow><mi mathvariant="script">H</mi></mrow><mo>≥</mo></msup><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow></math> , <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mi>η</mi><mi>i</mi><mo>≤</mo></msubsup><mo>></mo><mn>0</mn></mrow></math> , <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>i</mi><mo>∈</mo><msup><mrow><mi mathvariant="script">H</mi></mrow><mo>≤</mo></msup><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mrow></math> , <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>ν</mi><mi>i</mi></msub><mo>></mo><mn>0</mn></mrow></math> , <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>i</mi><mo>∈</mo><mi mathvariant="script">N</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></math> , and <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>μ</mi><mn>3</mn></msub><mo>></mo><mn>0</mn></mrow></math> if <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></msubsup><msub><mi>y</mi><mi>i</mi></msub><mo>=</mo><mi>n</mi><mo>-</mo><mi>s</mi></mrow></math> ,

ND3: the matrix <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mi>D</mi><mn>2</mn></msup><mspace width="0.166667em" /><msup><mi>L</mi><mi mathvariant="script">S</mi></msup><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><msub><mo>↾</mo><msubsup><mrow><mi mathvariant="script">T</mi></mrow><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><mi mathvariant="script">S</mi></msubsup></msub></mrow></math> is nonsingular.

Definition 12

(Quadratic index) Let (x, y) be a Karush–Kuhn–Tucker point of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="script">S</mi></math> with unique multipliers <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><mi>λ</mi><mo>,</mo><mi>μ</mi><mo>,</mo><mi>η</mi><mo>,</mo><mi>ν</mi><mo stretchy="false">)</mo></mrow></math> . The number of negative eigenvalues of the matrix <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mi>D</mi><mn>2</mn></msup><mspace width="0.166667em" /><msup><mi>L</mi><mi mathvariant="script">S</mi></msup><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><msub><mo>↾</mo><msubsup><mrow><mi mathvariant="script">T</mi></mrow><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><mi mathvariant="script">S</mi></msubsup></msub></mrow></math> is called its quadratic index (QI).

Note that ND1-ND3 are usual assumptions in nonlinear programming. ND1 refers to the linear independence constraint qualification, ND2 means the strict complementarity, and ND3 describes the second-order regularity. For the index of a nondegenerate Karush–Kuhn–Tucker point just the quadratic part is essential.

Lemma 2 examines the structure of y-components of a Karush–Kuhn–Tucker point of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="script">S</mi></math> .

Lemma 2

(Auxiliary y-variables in <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="script">S</mi></math> ) Let (x, y) be a Karush–Kuhn–Tucker point of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="script">S</mi></math> . Then, it holds:

the summation inequality constraint is active, i.e. <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></msubsup><msub><mi>y</mi><mi>i</mi></msub><mo>=</mo><mi>n</mi><mo>-</mo><mi>s</mi></mrow></math> ,

the index set <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="script">E</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo>∪</mo><mi mathvariant="script">H</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></math> consists of at least <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>n</mi><mo>-</mo><mi>s</mi><mo>-</mo><mn>1</mn></mrow></math> elements, and the index set <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="script">N</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></math> consists of at most s elements. Additionally, there is at most one index, that does not belong to any of these sets, i.e. <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mfenced close="|" open="|"><mi mathvariant="script">O</mi><mfenced close=")" open="("><mi>x</mi><mo>,</mo><mi>y</mi></mfenced></mfenced><mo>≤</mo><mn>1</mn></mrow></math> .

Proof

Let (x, y) be a Karush–Kuhn–Tucker point of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="script">S</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></msubsup><msub><mi>y</mi><mi>i</mi></msub><mo>></mo><mi>n</mi><mo>-</mo><mi>s</mi></mrow></math> . Then, there exist multipliers <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><mi>λ</mi><mo>,</mo><mi>μ</mi><mo>,</mo><mi>η</mi><mo>,</mo><mi>ν</mi><mo stretchy="false">)</mo></mrow></math> , such that (9)–(11) are fulfilled. Since <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>μ</mi><mn>3</mn></msub><mo>=</mo><mn>0</mn></mrow></math> , we have that the <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mi>i</mi><mo stretchy="false">)</mo></mrow></math> -th row of (9) reads as

Graph

Due to (10), (11), and <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>c</mi><mo>></mo><mn>0</mn></mrow></math> , it must hold that <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>i</mi><mo>∈</mo><mi mathvariant="script">N</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></math> for all <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>i</mi><mo>∈</mo><mfenced close="}" open="{"><mn>1</mn><mo>,</mo><mo>...</mo><mo>,</mo><mi>n</mi></mfenced></mrow></math> . This, however, contradicts <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></msubsup><msub><mi>y</mi><mi>i</mi></msub><mo>></mo><mi>n</mi><mo>-</mo><mi>s</mi></mrow></math> .

As in the proof of statement a), we conclude that <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>μ</mi><mn>3</mn></msub><mo>></mo><mn>0</mn></mrow></math> for a Karush–Kuhn–Tucker point (x, y) of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="script">S</mi></math> . Hence, the <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mi>i</mi><mo stretchy="false">)</mo></mrow></math> -th row now reads as

12 <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtable><mtr><mtd columnalign="right"><mrow><msub><mi>c</mi><mi>i</mi></msub><mo>=</mo><mfenced open="{"><mrow><mtable><mtr><mtd columnalign="left"><mrow><mo>-</mo><msub><mi>μ</mi><mrow><mn>2</mn><mo>,</mo><mi>i</mi></mrow></msub><mo>+</mo><msub><mi>μ</mi><mn>3</mn></msub><mo>,</mo></mrow></mtd><mtd columnalign="left"><mrow><mrow /><mspace width="0.333333em" /><mtext>for</mtext><mspace width="0.333333em" /><mi>i</mi><mo>∈</mo><mi mathvariant="script">E</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo stretchy="true">\</mo><mi mathvariant="script">H</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>,</mo></mrow></mtd></mtr><mtr><mtd columnalign="left"><mrow><mrow /><mo>-</mo><msub><mi>μ</mi><mrow><mn>2</mn><mo>,</mo><mi>i</mi></mrow></msub><mo>+</mo><msub><mi>μ</mi><mn>3</mn></msub><mo>+</mo><msubsup><mi>η</mi><mi>i</mi><mo>≥</mo></msubsup><msub><mi>x</mi><mi>i</mi></msub><mo>,</mo></mrow></mtd><mtd columnalign="left"><mrow><mrow /><mspace width="0.333333em" /><mtext>for</mtext><mspace width="0.333333em" /><mi>i</mi><mo>∈</mo><mi mathvariant="script">E</mi><mrow><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><mo>∩</mo><msup><mrow><mi mathvariant="script">H</mi></mrow><mo>≥</mo></msup><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><mo>,</mo></mrow></mtd></mtr><mtr><mtd columnalign="left"><mrow><mrow /><mo>-</mo><msub><mi>μ</mi><mrow><mn>2</mn><mo>,</mo><mi>i</mi></mrow></msub><mo>+</mo><msub><mi>μ</mi><mn>3</mn></msub><mo>-</mo><msubsup><mi>η</mi><mi>i</mi><mo>≤</mo></msubsup><msub><mi>x</mi><mi>i</mi></msub><mo>,</mo></mrow></mtd><mtd columnalign="left"><mrow><mrow /><mspace width="0.333333em" /><mtext>for</mtext><mspace width="0.333333em" /><mi>i</mi><mo>∈</mo><mi mathvariant="script">E</mi><mrow><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><mo>∩</mo><msup><mrow><mi mathvariant="script">H</mi></mrow><mo>≤</mo></msup><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><mo>,</mo></mrow></mtd></mtr><mtr><mtd columnalign="left"><mrow><mrow /><msub><mi>μ</mi><mn>3</mn></msub><mo>+</mo><msubsup><mi>η</mi><mi>i</mi><mo>≥</mo></msubsup><msub><mi>x</mi><mi>i</mi></msub><mo>,</mo></mrow></mtd><mtd columnalign="left"><mrow><mrow /><mspace width="0.333333em" /><mtext>for</mtext><mspace width="0.333333em" /><mi>i</mi><mo>∈</mo><msup><mrow><mi mathvariant="script">H</mi></mrow><mo>≥</mo></msup><mrow><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><mo stretchy="true">\</mo><mi mathvariant="script">E</mi><mrow><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><mo>,</mo></mrow></mrow></mtd></mtr><mtr><mtd columnalign="left"><mrow><mrow /><msub><mi>μ</mi><mn>3</mn></msub><mo>-</mo><msubsup><mi>η</mi><mi>i</mi><mo>≤</mo></msubsup><msub><mi>x</mi><mi>i</mi></msub><mo>,</mo></mrow></mtd><mtd columnalign="left"><mrow><mrow /><mspace width="0.333333em" /><mtext>for</mtext><mspace width="0.333333em" /><mi>i</mi><mo>∈</mo><msup><mrow><mi mathvariant="script">H</mi></mrow><mo>≤</mo></msup><mrow><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><mo stretchy="true">\</mo><mi mathvariant="script">E</mi><mrow><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><mo>,</mo></mrow></mrow></mtd></mtr><mtr><mtd columnalign="left"><mrow><mrow /><msub><mi>μ</mi><mn>3</mn></msub><mo>+</mo><msub><mi>ν</mi><mi>i</mi></msub><mo>,</mo></mrow></mtd><mtd columnalign="left"><mrow><mrow /><mspace width="0.333333em" /><mtext>for</mtext><mspace width="0.333333em" /><mi>i</mi><mo>∈</mo><mi mathvariant="script">N</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo>,</mo></mrow></mtd></mtr><mtr><mtd columnalign="left"><mrow><mrow /><msub><mi>μ</mi><mn>3</mn></msub><mo>,</mo></mrow></mtd><mtd columnalign="left"><mrow><mrow /><mspace width="0.333333em" /><mtext>else</mtext><mspace width="0.333333em" /><mo>.</mo></mrow></mtd></mtr></mtable></mrow></mfenced></mrow></mtd></mtr></mtable></mrow></math>

Graph

It follows from (12) and the components of c being pairwise different that there can be at most one element <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mover accent="true"><mrow><mi>i</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>∈</mo><mi mathvariant="script">O</mi><mfenced close=")" open="("><mi>x</mi><mo>,</mo><mi>y</mi></mfenced></mrow></math> . If <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="script">E</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo>∪</mo><mi mathvariant="script">H</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></math> consists of fewer than <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>n</mi><mo>-</mo><mi>s</mi><mo>-</mo><mn>1</mn></mrow></math> elements, we get:

Graph

a contradiction. Finally, we assume that <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="script">N</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></math> consists of more than s elements. In this case, there are at most <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>n</mi><mo>-</mo><mi>s</mi><mo>-</mo><mn>1</mn></mrow></math> nonvanishing components of y. Consequently,

Graph

provides a contradiction.

We apply the general result on the Scholtes-type regularization of MPOC in our context for the regularized continuous reformulation <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="script">R</mi></math> , see [[4]].

Theorem 3

(Convergence from <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="script">S</mi></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="script">R</mi></math> , cf. [[4]]) Suppose that a sequence of Karush–Kuhn–Tucker points <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><msup><mi>x</mi><mi>t</mi></msup><mo>,</mo><msup><mi>y</mi><mi>t</mi></msup><mo stretchy="false">)</mo></mrow></math> of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="script">S</mi></math> converges to <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close=")" open="("><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mfenced></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>t</mi><mo stretchy="false">→</mo><mn>0</mn></mrow></math> . If MPOC-LICQ holds at <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close=")" open="("><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mfenced></math> , then it is a T-stationary point of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="script">R</mi></math> .

From the proof of Theorem 3 in [[4]] also the convergence of the corresponding multipliers can be deduced.

Remark 1

(Convergence of multipliers) Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close=")" open="("><msup><mi>λ</mi><mi>t</mi></msup><mo>,</mo><msup><mi>μ</mi><mi>t</mi></msup><mo>,</mo><msup><mi>η</mi><mi>t</mi></msup><mo>,</mo><msup><mi>ν</mi><mi>t</mi></msup></mfenced></math> be the multipliers of the Karush–Kuhn–Tucker points <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close=")" open="("><msup><mi>x</mi><mi>t</mi></msup><mo>,</mo><msup><mi>y</mi><mi>t</mi></msup></mfenced></math> of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="script">S</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close=")" open="("><mover accent="true"><mrow><mi>λ</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>μ</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>σ</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>ϱ</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mfenced></math> of the T-stationary point <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close=")" open="("><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mfenced></math> of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="script">R</mi></math> as in Theorem 3. Due to MPOC-LICQ at <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close=")" open="("><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mfenced></math> , we have:

<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><munder><mo movablelimits="false">lim</mo><mrow><mi>t</mi><mo stretchy="false">→</mo><mn>0</mn></mrow></munder><msup><mi>λ</mi><mi>t</mi></msup><mo>=</mo><mover accent="true"><mrow><mi>λ</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mrow></math> , <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><munder><mo movablelimits="false">lim</mo><mrow><mi>t</mi><mo stretchy="false">→</mo><mn>0</mn></mrow></munder><msup><mi>μ</mi><mi>t</mi></msup><mo>=</mo><mover accent="true"><mrow><mi>μ</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mrow></math> ,

<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><munder><mo movablelimits="false">lim</mo><mrow><mi>t</mi><mo stretchy="false">→</mo><mn>0</mn></mrow></munder><msubsup><mi>ν</mi><mrow><mi>i</mi></mrow><mi>t</mi></msubsup><mo>+</mo><mfenced close=")" open="("><msubsup><mi>η</mi><mrow><mi>i</mi></mrow><mrow><mo>≥</mo><mo>,</mo><mi>t</mi></mrow></msubsup><mo>-</mo><msubsup><mi>η</mi><mrow><mi>i</mi></mrow><mrow><mo>≤</mo><mo>,</mo><mi>t</mi></mrow></msubsup></mfenced><msubsup><mi>x</mi><mrow><mi>i</mi></mrow><mi>t</mi></msubsup><mo>=</mo><msub><mover accent="true"><mrow><mi>σ</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mrow><mn>2</mn><mo>,</mo><mi>i</mi></mrow></msub></mrow></math> , <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>i</mi><mo>∈</mo><msub><mi>a</mi><mn>10</mn></msub><mfenced close=")" open="("><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mfenced></mrow></math> ,

<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><munder><mo movablelimits="false">lim</mo><mrow><mi>t</mi><mo stretchy="false">→</mo><mn>0</mn></mrow></munder><mfenced close=")" open="("><msubsup><mi>η</mi><mrow><mi>i</mi></mrow><mrow><mo>≥</mo><mo>,</mo><mi>t</mi></mrow></msubsup><mo>-</mo><msubsup><mi>η</mi><mrow><mi>i</mi></mrow><mrow><mo>≤</mo><mo>,</mo><mi>t</mi></mrow></msubsup></mfenced><msubsup><mi>y</mi><mrow><mi>i</mi></mrow><mi>t</mi></msubsup><mo>=</mo><msub><mover accent="true"><mrow><mi>ϱ</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mrow><mn>1</mn><mo>,</mo><mi>i</mi></mrow></msub></mrow></math> , <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><munder><mo movablelimits="false">lim</mo><mrow><mi>t</mi><mo stretchy="false">→</mo><mn>0</mn></mrow></munder><msubsup><mi>ν</mi><mrow><mi>i</mi></mrow><mi>t</mi></msubsup><mo>+</mo><mfenced close=")" open="("><msubsup><mi>η</mi><mrow><mi>i</mi></mrow><mrow><mo>≥</mo><mo>,</mo><mi>t</mi></mrow></msubsup><mo>-</mo><msubsup><mi>η</mi><mrow><mi>i</mi></mrow><mrow><mo>≤</mo><mo>,</mo><mi>t</mi></mrow></msubsup></mfenced><msubsup><mi>x</mi><mrow><mi>i</mi></mrow><mi>t</mi></msubsup><mo>=</mo><msub><mover accent="true"><mrow><mi>ϱ</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mrow><mn>2</mn><mo>,</mo><mi>i</mi></mrow></msub></mrow></math> , <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>i</mi><mo>∈</mo><msub><mi>a</mi><mn>00</mn></msub><mfenced close=")" open="("><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mfenced></mrow></math> .

The convergence of nondegenerate Karush–Kuhn–Tucker points of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="script">S</mi></math> does not prevent the limiting T-stationary point of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="script">S</mi></math> from being degenerate. Let us present in Example 1 the failure of NDT2. Examples with the failure of NDT1, NDT3, or NDT4 are not difficult to construct analogously.

Example 1

(Failure of NDT2) We consider the following Scholtes-type regularization <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="script">S</mi></math> with <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>n</mi><mo>=</mo><mn>2</mn></mrow></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>s</mi><mo>=</mo><mn>1</mn></mrow></math> :

Graph

as well as the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><mo stretchy="false">(</mo><msup><mi>x</mi><mi>t</mi></msup><mo>,</mo><msup><mi>y</mi><mi>t</mi></msup><mo stretchy="false">)</mo></mrow><mo>=</mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo>,</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></mrow></math> .We claim that this point is a nondegenerate Karush–Kuhn–Tucker point for <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>t</mi><mo><</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>-</mo><msqrt><mfrac><mn>13</mn><mn>72</mn></mfrac></msqrt></mrow></math> . Indeed, it holds:

Graph

with the positive multipliers <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mi>μ</mi><mn>3</mn><mi>t</mi></msubsup><mo>=</mo><msub><mi>c</mi><mn>1</mn></msub><mo>+</mo><mn>2</mn><mi>t</mi><mo>-</mo><mn>2</mn><msup><mi>t</mi><mn>2</mn></msup></mrow></math> , <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mi>η</mi><mn>1</mn><mrow><mo>≤</mo><mo>,</mo><mi>t</mi></mrow></msubsup><mo>=</mo><mn>2</mn><mo>-</mo><mn>2</mn><mi>t</mi></mrow></math> , <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mi>ν</mi><mn>2</mn><mi>t</mi></msubsup><mo>=</mo><mfrac><mn>5</mn><mn>36</mn></mfrac><mo>-</mo><mn>2</mn><mi>t</mi><mo>+</mo><mn>2</mn><msup><mi>t</mi><mn>2</mn></msup></mrow></math> . The tangent space is <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mrow><mi mathvariant="script">T</mi></mrow><mrow><mo stretchy="false">(</mo><msup><mi>x</mi><mi>t</mi></msup><mo>,</mo><msup><mi>y</mi><mi>t</mi></msup><mo stretchy="false">)</mo></mrow><mi mathvariant="script">S</mi></msubsup><mo>=</mo><mfenced close="}" open="{"><mi>ξ</mi><mo>∈</mo><msup><mrow><mi mathvariant="double-struck">R</mi></mrow><mn>4</mn></msup><mspace width="0.166667em" /><mfenced open="|"><mspace width="0.166667em" /><msub><mi>ξ</mi><mn>1</mn></msub><mo>=</mo><msub><mi>ξ</mi><mn>3</mn></msub><mo>=</mo><msub><mi>ξ</mi><mn>4</mn></msub><mo>=</mo><mn>0</mn></mfenced></mfenced></mrow></math> . The Hessian of the corresponding Lagrange function is

Graph

Therefore, it is straightforward that <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mi>D</mi><mn>2</mn></msup><mspace width="0.166667em" /><msup><mi>L</mi><mi mathvariant="script">S</mi></msup><mrow><mo stretchy="false">(</mo><msup><mi>x</mi><mi>t</mi></msup><mo>,</mo><msup><mi>y</mi><mi>t</mi></msup><mo stretchy="false">)</mo></mrow><msub><mo>↾</mo><msubsup><mrow><mi mathvariant="script">T</mi></mrow><mrow><mo stretchy="false">(</mo><msup><mi>x</mi><mi>t</mi></msup><mo>,</mo><msup><mi>y</mi><mi>t</mi></msup><mo stretchy="false">)</mo></mrow><mi mathvariant="script">S</mi></msubsup></msub></mrow></math> is nonsingular. We conclude that ND1-ND3 are fulfilled at <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><msup><mi>x</mi><mi>t</mi></msup><mo>,</mo><msup><mi>y</mi><mi>t</mi></msup><mo stretchy="false">)</mo></mrow></math> . Moreover, <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><msup><mi>x</mi><mi>t</mi></msup><mo>,</mo><msup><mi>y</mi><mi>t</mi></msup><mo stretchy="false">)</mo></mrow></math> converges to <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></math> if <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>t</mi><mo stretchy="false">→</mo><mn>0</mn></mrow></math> . This point is T-stationary for the corresponding regularized continuous reformulation <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="script">R</mi></math> according to Theorem 3, since MPOC-LICQ is fulfilled. Indeed, we obtain the T-stationarity condition

Graph

Due to Example 1, we cannot expect that a T-stationary point of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="script">R</mi></math> , which is the limit of a sequence of nondegenerate Karush–Kuhn–Tucker points of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="script">S</mi></math> , is also nondegenerate. Instead, we intend to examine its type if assuming nondegeneracy. Next Lemma 3 provides some valuable insights into the relations between active index sets while doing so.

Lemma 3

(Active index sets) Suppose a sequence of Karush–Kuhn–Tucker points <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close=")" open="("><msup><mi>x</mi><mi>t</mi></msup><mo>,</mo><msup><mi>y</mi><mi>t</mi></msup></mfenced></math> of <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="script">S</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></math> converges to <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close=")" open="("><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mfenced></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>t</mi><mo stretchy="false">→</mo><mn>0</mn></mrow></math> . Moreover, let <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo stretchy="false">)</mo></mrow></math> be a nondegenerate T-stationary point of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="script">R</mi></math> . Then, for all sufficiently small t it holds:

<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>Q</mi><mn>0</mn></msub><mfenced close=")" open="("><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mfenced><mo>=</mo><msub><mi>Q</mi><mn>0</mn></msub><mfenced close=")" open="("><msup><mi>x</mi><mi>t</mi></msup></mfenced></mrow></math> ,

<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="script">E</mi><mfenced close=")" open="("><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mfenced><mo>=</mo><mi mathvariant="script">E</mi><mfenced close=")" open="("><msup><mi>y</mi><mi>t</mi></msup></mfenced></mrow></math> ,

<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>a</mi><mn>00</mn></msub><mfenced close=")" open="("><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mfenced><mo>⊂</mo><mi mathvariant="script">H</mi><mfenced close=")" open="("><msup><mi>x</mi><mi>t</mi></msup><mo>,</mo><msup><mi>y</mi><mi>t</mi></msup></mfenced></mrow></math> ,

<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="script">N</mi><mfenced close=")" open="("><msup><mi>y</mi><mi>t</mi></msup></mfenced><mo>⊂</mo><msub><mi>a</mi><mn>10</mn></msub><mfenced close=")" open="("><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mfenced><mo>⊂</mo><mi mathvariant="script">N</mi><mfenced close=")" open="("><msup><mi>y</mi><mi>t</mi></msup></mfenced><mo>∪</mo><mi mathvariant="script">H</mi><mfenced close=")" open="("><msup><mi>x</mi><mi>t</mi></msup><mo>,</mo><msup><mi>y</mi><mi>t</mi></msup></mfenced></mrow></math> .

Proof

a) We start by proving <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>Q</mi><mn>0</mn></msub><mfenced close=")" open="("><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mfenced><mo>=</mo><msub><mi>Q</mi><mn>0</mn></msub><mfenced close=")" open="("><msup><mi>x</mi><mi>t</mi></msup></mfenced></mrow></math> . Due to continuity arguments, we have <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>Q</mi><mn>0</mn></msub><mfenced close=")" open="("><msup><mi>x</mi><mi>t</mi></msup></mfenced><mo>⊂</mo><msub><mi>Q</mi><mn>0</mn></msub><mfenced close=")" open="("><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mfenced></mrow></math> for all sufficiently small t. Let us now assume that there exists <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mover accent="true"><mrow><mi>i</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>∈</mo><msub><mi>Q</mi><mn>0</mn></msub><mfenced close=")" open="("><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mfenced><mrow><mo stretchy="true">\</mo></mrow><msub><mi>Q</mi><mn>0</mn></msub><mfenced close=")" open="("><msup><mi>x</mi><mi>t</mi></msup></mfenced></mrow></math> along a subsequence. Hence, for the corresponding multipliers it holds <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mi>μ</mi><mrow><mover accent="true"><mrow><mi>i</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mrow><mi>t</mi></msubsup><mo>=</mo><mn>0</mn></mrow></math> . NDT1 allows us to apply Remark 1, and we thus have <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mover accent="true"><mrow><mi>μ</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mover accent="true"><mrow><mi>i</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></msub><mo>=</mo><munder><mo movablelimits="false">lim</mo><mrow><mi>t</mi><mo stretchy="false">→</mo><mi>∞</mi></mrow></munder><msubsup><mi>μ</mi><mrow><mover accent="true"><mrow><mi>i</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mrow><mi>t</mi></msubsup><mo>=</mo><mn>0</mn></mrow></math> , a contradiction to NDT2. Consequently, <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>Q</mi><mn>0</mn></msub><mfenced close=")" open="("><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mfenced><mo>=</mo><msub><mi>Q</mi><mn>0</mn></msub><mfenced close=")" open="("><msup><mi>x</mi><mi>t</mi></msup></mfenced></mrow></math> holds for all sufficiently small t.

b) Next, we prove <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="script">E</mi><mfenced close=")" open="("><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mfenced><mo>=</mo><mi mathvariant="script">E</mi><mfenced close=")" open="("><msup><mi>y</mi><mi>t</mi></msup></mfenced></mrow></math> . Again, continuity arguments provide <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="script">E</mi><mfenced close=")" open="("><msup><mi>y</mi><mi>t</mi></msup></mfenced><mo>⊂</mo><mi mathvariant="script">E</mi><mfenced close=")" open="("><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mfenced></mrow></math> for all sufficiently small t. Similar to the first part of the proof, we now assume there exists <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mover accent="true"><mrow><mi>i</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>∈</mo><mi mathvariant="script">E</mi><mfenced close=")" open="("><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mfenced><mrow><mo stretchy="true">\</mo><mi mathvariant="script">E</mi></mrow><mfenced close=")" open="("><msup><mi>y</mi><mi>t</mi></msup></mfenced></mrow></math> along a subsequence. As we have seen in Lemma 1, T-stationarity of <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo stretchy="false">)</mo></mrow></math> implies in particular <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>c</mi><mover accent="true"><mrow><mi>i</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></msub><mo>=</mo><mo>-</mo><msub><mover accent="true"><mrow><mi>μ</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mrow><mn>2</mn><mo>,</mo><mover accent="true"><mrow><mi>i</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mrow></msub><mo>+</mo><msub><mover accent="true"><mrow><mi>μ</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mrow><mn>3</mn><mo>,</mo><mover accent="true"><mrow><mi>i</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mrow></msub></mrow></math> . Moreover, NDT1 and Remark 1 provide <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><munder><mo movablelimits="false">lim</mo><mrow><mi>t</mi><mo stretchy="false">→</mo><mn>0</mn></mrow></munder><msubsup><mi>μ</mi><mrow><mn>3</mn></mrow><mi>t</mi></msubsup><mo>=</mo><msub><mover accent="true"><mrow><mi>μ</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mn>3</mn></msub></mrow></math> . Since <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mover accent="true"><mrow><mi>i</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>∉</mo><mi mathvariant="script">N</mi><mfenced close=")" open="("><msup><mi>y</mi><mi>t</mi></msup></mfenced></mrow></math> , we distinguish the following cases:

(i)

. Karush–Kuhn–Tucker conditions for

• imply

, cf. (12). It follows

. By taking the limit, we can cancel out

• and

. This leads to a contradiction because the left-hand side of the equation is strictly negative due to NDT2 and the right-hand side is nonnegative since

is nonnegative and

is positive.

(ii)

. By using (12), we get

. This leads to a contradiction just as in the previous case.

(iii)

. Analogously, we obtain

from (12). It follows

. Taking the limits leads to

, a contradiction with NDT2.

Altogether, <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="script">E</mi><mfenced close=")" open="("><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mfenced><mrow><mo stretchy="true">\</mo><mi mathvariant="script">E</mi></mrow><mfenced close=")" open="("><msup><mi>y</mi><mi>t</mi></msup></mfenced><mo>=</mo><mi mathvariant="normal">∅</mi></mrow></math> for all sufficiently small t, and the assertion follows.

c) Clearly, <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>a</mi><mn>00</mn></msub><mfenced close=")" open="("><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mfenced><mo>∩</mo><mi mathvariant="script">E</mi><mrow><mo stretchy="false">(</mo><msup><mi>y</mi><mi>t</mi></msup><mo stretchy="false">)</mo></mrow><mo>=</mo><mi mathvariant="normal">∅</mi></mrow></math> for sufficiently small t.

Let us assume there exists an <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mover accent="true"><mrow><mi>i</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>∈</mo><msub><mi>a</mi><mn>00</mn></msub><mrow><mo stretchy="false">(</mo><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo stretchy="false">)</mo></mrow><mo>∩</mo><mi mathvariant="script">N</mi><mfenced close=")" open="("><msup><mi>y</mi><mi>t</mi></msup></mfenced></mrow></math> . In view of (12), we have <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>c</mi><mover accent="true"><mrow><mi>i</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></msub><mo>=</mo><msubsup><mi>μ</mi><mn>3</mn><mi>t</mi></msubsup><mo>+</mo><msubsup><mi>ν</mi><mrow><mover accent="true"><mrow><mi>i</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mrow><mi>t</mi></msubsup></mrow></math> . Due to the T-stationarity of <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close=")" open="("><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mfenced></math> , the <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mi>i</mi><mo stretchy="false">)</mo></mrow></math> -th row of (5) reads as

13 <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtable><mtr><mtd columnalign="right"><mrow><msub><mi>c</mi><mi>i</mi></msub><mo>=</mo><mfenced open="{"><mrow><mtable><mtr><mtd columnalign="left"><mrow><mo>-</mo><msub><mover accent="true"><mrow><mi>μ</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mrow><mn>2</mn><mo>,</mo><mi>i</mi></mrow></msub><mo>+</mo><msub><mover accent="true"><mrow><mi>μ</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mn>3</mn></msub><mo>,</mo></mrow></mtd><mtd columnalign="left"><mrow><mrow /><mspace width="0.333333em" /><mtext>for</mtext><mspace width="0.333333em" /><mi>i</mi><mo>∈</mo><mi mathvariant="script">E</mi><mo stretchy="false">(</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo stretchy="false">)</mo><mo>,</mo></mrow></mtd></mtr><mtr><mtd columnalign="left"><mrow><mrow /><msub><mover accent="true"><mrow><mi>σ</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mrow><mn>2</mn><mo>,</mo><mi>i</mi></mrow></msub><mo>+</mo><msub><mover accent="true"><mrow><mi>μ</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mn>3</mn></msub><mo>,</mo></mrow></mtd><mtd columnalign="left"><mrow><mrow /><mspace width="0.333333em" /><mtext>for</mtext><mspace width="0.333333em" /><mi>i</mi><mo>∈</mo><msub><mi>a</mi><mn>10</mn></msub><mfenced close=")" open="("><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mfenced><mo>,</mo></mrow></mtd></mtr><mtr><mtd columnalign="left"><mrow><mrow /><msub><mover accent="true"><mrow><mi>ϱ</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mrow><mn>2</mn><mo>,</mo><mi>i</mi></mrow></msub><mo>+</mo><msub><mover accent="true"><mrow><mi>μ</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mn>3</mn></msub><mo>,</mo></mrow></mtd><mtd columnalign="left"><mrow><mrow /><mspace width="0.333333em" /><mtext>for</mtext><mspace width="0.333333em" /><mi>i</mi><mo>∈</mo><msub><mi>a</mi><mn>00</mn></msub><mfenced close=")" open="("><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mfenced><mo>,</mo></mrow></mtd></mtr><mtr><mtd columnalign="left"><mrow><mrow /><msub><mover accent="true"><mrow><mi>μ</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mn>3</mn></msub><mo>,</mo></mrow></mtd><mtd columnalign="left"><mrow><mrow /><mspace width="0.333333em" /><mtext>else</mtext><mspace width="0.333333em" /><mo>.</mo></mrow></mtd></mtr></mtable></mrow></mfenced></mrow></mtd></mtr></mtable></mrow></math>

Graph

This provides <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>c</mi><mover accent="true"><mrow><mi>i</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></msub><mo>=</mo><msub><mover accent="true"><mrow><mi>ϱ</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mrow><mn>2</mn><mo>,</mo><mover accent="true"><mrow><mi>i</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mrow></msub><mo>+</mo><msub><mover accent="true"><mrow><mi>μ</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mn>3</mn></msub></mrow></math> . According to Remark 1, we have <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><munder><mo movablelimits="false">lim</mo><mrow><mi>t</mi><mo stretchy="false">→</mo><mn>0</mn></mrow></munder><msubsup><mi>μ</mi><mrow><mn>3</mn></mrow><mi>t</mi></msubsup><mo>=</mo><msub><mover accent="true"><mrow><mi>μ</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mn>3</mn></msub></mrow></math> . Consequently, it must hold <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><munder><mo movablelimits="false">lim</mo><mrow><mi>t</mi><mo stretchy="false">→</mo><mn>0</mn></mrow></munder><msubsup><mi>ν</mi><mrow><mover accent="true"><mrow><mi>i</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mrow><mi>t</mi></msubsup><mo>=</mo><msub><mover accent="true"><mrow><mi>ϱ</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mrow><mn>2</mn><mo>,</mo><mover accent="true"><mrow><mi>i</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mrow></msub></mrow></math> . This, however, cannot be true since <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mi>ν</mi><mrow><mover accent="true"><mrow><mi>i</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mrow><mi>t</mi></msubsup><mo>≥</mo><mn>0</mn></mrow></math> , while <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mover accent="true"><mrow><mi>ϱ</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mrow><mn>2</mn><mo>,</mo><mover accent="true"><mrow><mi>i</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mrow></msub><mo><</mo><mn>0</mn></mrow></math> due to NDT3 from the nondegeneracy of <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo stretchy="false">)</mo></mrow></math> , a contradiction. Let us assume now that there exists an <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mover accent="true"><mrow><mi>i</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>∈</mo><msub><mi>a</mi><mn>00</mn></msub><mrow><mo stretchy="false">(</mo><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo stretchy="false">)</mo></mrow><mo>∩</mo><mi mathvariant="script">O</mi><mfenced close=")" open="("><msup><mi>x</mi><mi>t</mi></msup><mo>,</mo><msup><mi>y</mi><mi>t</mi></msup></mfenced></mrow></math> . Analogously, we get <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mover accent="true"><mrow><mi>ϱ</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mrow><mn>2</mn><mo>,</mo><mover accent="true"><mrow><mi>i</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mrow></msub><mo>=</mo><mn>0</mn></mrow></math> , again a contradiction to NDT3. Overall, we get the assertion.

d) Clearly, <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>a</mi><mn>01</mn></msub><mfenced close=")" open="("><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mfenced><mo>∩</mo><mi mathvariant="script">N</mi><mrow><mo stretchy="false">(</mo><msup><mi>y</mi><mi>t</mi></msup><mo stretchy="false">)</mo></mrow><mo>=</mo><mi mathvariant="normal">∅</mi></mrow></math> for sufficiently small t. From c) we also know that <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>a</mi><mn>00</mn></msub><mfenced close=")" open="("><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mfenced><mo>∩</mo><mi mathvariant="script">N</mi><mrow><mo stretchy="false">(</mo><msup><mi>y</mi><mi>t</mi></msup><mo stretchy="false">)</mo></mrow><mo>=</mo><mi mathvariant="normal">∅</mi></mrow></math> . Altogether, the first inclusion of the assertion follows immediately. Further, it also holds <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>a</mi><mn>10</mn></msub><mfenced close=")" open="("><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mfenced><mo>∩</mo><mi mathvariant="script">E</mi><mrow><mo stretchy="false">(</mo><msup><mi>y</mi><mi>t</mi></msup><mo stretchy="false">)</mo></mrow><mo>=</mo><mi mathvariant="normal">∅</mi></mrow></math> for sufficiently small t. Let us assume there exists an <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mover accent="true"><mrow><mi>i</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>∈</mo><msub><mi>a</mi><mn>10</mn></msub><mrow><mo stretchy="false">(</mo><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo stretchy="false">)</mo></mrow><mo>∩</mo><mi mathvariant="script">O</mi><mfenced close=")" open="("><msup><mi>x</mi><mi>t</mi></msup><mo>,</mo><msup><mi>y</mi><mi>t</mi></msup></mfenced></mrow></math> . Due to (12), we have <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>c</mi><mover accent="true"><mrow><mi>i</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></msub><mo>=</mo><msubsup><mi>μ</mi><mn>3</mn><mi>t</mi></msubsup></mrow></math> . In view of Lemma 1c), there exists an index <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mover accent="true"><mi>i</mi><mo stretchy="false">~</mo></mover><mo>∈</mo><msub><mi>a</mi><mn>01</mn></msub><mrow><mrow><mo stretchy="false">(</mo><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo stretchy="false">)</mo></mrow><mo stretchy="true">\</mo><mi mathvariant="script">E</mi></mrow><mfenced close=")" open="("><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mfenced></mrow></math> . Thus, T-stationarity of <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo stretchy="false">)</mo></mrow></math> implies via (13) that <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>c</mi><mover accent="true"><mi>i</mi><mo stretchy="false">~</mo></mover></msub><mo>=</mo><msub><mover accent="true"><mrow><mi>μ</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mn>3</mn></msub></mrow></math> . By taking the limit and Remark 1, we obtain <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>c</mi><mover accent="true"><mrow><mi>i</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></msub><mo>=</mo><msub><mi>c</mi><mover accent="true"><mi>i</mi><mo stretchy="false">~</mo></mover></msub></mrow></math> , but <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mover accent="true"><mrow><mi>i</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>≠</mo><mover accent="true"><mi>i</mi><mo stretchy="false">~</mo></mover></mrow></math> , a contradiction to the choice of c. <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>□</mo></math>

Theorem 4 highlights the convergence properties of the Scholtes-type regularization method. Its proof can be found in the Appendix below.

Theorem 4

(Convergence from <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="script">S</mi></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="script">R</mi></math> again) Suppose that a sequence of nondegenerate Karush–Kuhn–Tucker points <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><msup><mi>x</mi><mi>t</mi></msup><mo>,</mo><msup><mi>y</mi><mi>t</mi></msup><mo stretchy="false">)</mo></mrow></math> of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="script">S</mi></math> with quadratic index m converges to <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close=")" open="("><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mfenced></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>t</mi><mo stretchy="false">→</mo><mn>0</mn></mrow></math> . If <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo stretchy="false">)</mo></mrow></math> is a nondegenerate T-stationary point of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="script">R</mi></math> , then we have for its T-index:

Graph

If additionally NDT6 holds at <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo stretchy="false">)</mo></mrow></math> , then the indices coincide, i.e. <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>T</mi><mi>I</mi><mo>=</mo><mi>m</mi></mrow></math> .

Let us illustrate the necessity of NDT6 for the validity of Theorem 4.

Example 2

(Necessity of NDT6) We consider the following Scholtes-type regularization <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="script">S</mi></math> with <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>n</mi><mo>=</mo><mn>2</mn></mrow></math> , <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>s</mi><mo>=</mo><mn>1</mn></mrow></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mn>0</mn><mo><</mo><msub><mi>c</mi><mn>1</mn></msub><mo><</mo><msub><mi>c</mi><mn>2</mn></msub></mrow></math> :

Graph

as well as the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mfenced close=")" open="("><msup><mi>x</mi><mi>t</mi></msup><mo>,</mo><msup><mi>y</mi><mi>t</mi></msup></mfenced><mo>=</mo><mrow><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></mrow></math> . We claim that this point is a nondegenerate Karush–Kuhn–Tucker point. Indeed, it holds:

Graph

with the positive multipliers <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mi>μ</mi><mrow><mn>1</mn><mo>,</mo><mn>1</mn></mrow><mi>t</mi></msubsup><mo>=</mo><mn>2</mn><mo>,</mo><msubsup><mi>μ</mi><mn>3</mn><mi>t</mi></msubsup><mo>=</mo><msub><mi>c</mi><mn>1</mn></msub><mo>,</mo><msubsup><mi>ν</mi><mn>2</mn><mi>t</mi></msubsup><mo>=</mo><msub><mi>c</mi><mn>2</mn></msub><mo>-</mo><msub><mi>c</mi><mn>1</mn></msub></mrow></math> . Obviously, LICQ and strict complementarity, i.e. ND1 and ND2, respectively, are fulfilled. We show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mi>D</mi><mn>2</mn></msup><mspace width="0.166667em" /><msup><mi>L</mi><mi mathvariant="script">S</mi></msup><mrow><mo stretchy="false">(</mo><msup><mi>x</mi><mi>t</mi></msup><mo>,</mo><msup><mi>y</mi><mi>t</mi></msup><mo stretchy="false">)</mo></mrow><msub><mo>↾</mo><msubsup><mrow><mi mathvariant="script">T</mi></mrow><mrow><mo stretchy="false">(</mo><msup><mi>x</mi><mi>t</mi></msup><mo>,</mo><msup><mi>y</mi><mi>t</mi></msup><mo stretchy="false">)</mo></mrow><mi mathvariant="script">S</mi></msubsup></msub></mrow></math> is nonsingular and calculate the number of its negative eigenvalues. The tangent space is <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mrow><mi mathvariant="script">T</mi></mrow><mrow><mo stretchy="false">(</mo><msup><mi>x</mi><mi>t</mi></msup><mo>,</mo><msup><mi>y</mi><mi>t</mi></msup><mo stretchy="false">)</mo></mrow><mi mathvariant="script">S</mi></msubsup><mo>=</mo><mfenced close="}" open="{"><mi>ξ</mi><mo>∈</mo><msup><mrow><mi mathvariant="double-struck">R</mi></mrow><mn>4</mn></msup><mspace width="0.166667em" /><mfenced open="|"><mspace width="0.166667em" /><msub><mi>ξ</mi><mn>1</mn></msub><mo>+</mo><msub><mi>ξ</mi><mn>2</mn></msub><mo>=</mo><mn>0</mn><mo>,</mo><msub><mi>ξ</mi><mn>3</mn></msub><mo>=</mo><msub><mi>ξ</mi><mn>4</mn></msub><mo>=</mo><mn>0</mn></mfenced></mfenced></mrow></math> . For the Hessian of the corresponding Lagrange function we have:

Graph

Thus, for <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>ξ</mi><mo>∈</mo><msubsup><mrow><mi mathvariant="script">T</mi></mrow><mrow><mo stretchy="false">(</mo><msup><mi>x</mi><mi>t</mi></msup><mo>,</mo><msup><mi>y</mi><mi>t</mi></msup><mo stretchy="false">)</mo></mrow><mi mathvariant="script">S</mi></msubsup></mrow></math> it holds:

Graph

Hence, ND3 is also fulfilled, the Karush–Kuhn–Tucker point <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close=")" open="("><msup><mi>x</mi><mi>t</mi></msup><mo>,</mo><msup><mi>y</mi><mi>t</mi></msup></mfenced></math> is nondegenerate and its quadratic index equals one, i.e. <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>m</mi><mo>=</mo><mn>1</mn></mrow></math> in Theorem 4. The limiting point is <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></math> . This point is T-stationary for the corresponding regularized continuous reformulation <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="script">R</mi></math> according to Theorem 3, since MPOC-LICQ is fulfilled. Indeed, we have:

Graph

with the unique multipliers <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mover accent="true"><mrow><mi>μ</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mrow><mn>1</mn><mo>,</mo><mn>1</mn></mrow></msub><mo>=</mo><mn>2</mn><mo>,</mo><msub><mover accent="true"><mrow><mi>μ</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mn>3</mn></msub><mo>=</mo><msub><mi>c</mi><mn>1</mn></msub><mo>,</mo><msub><mover accent="true"><mrow><mi>σ</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mrow><mn>1</mn><mo>,</mo><mn>1</mn></mrow></msub><mo>=</mo><mn>0</mn><mo>,</mo><msub><mover accent="true"><mrow><mi>σ</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mrow><mn>2</mn><mo>,</mo><mn>2</mn></mrow></msub><mo>=</mo><msub><mi>c</mi><mn>2</mn></msub><mo>-</mo><msub><mi>c</mi><mn>1</mn></msub><mo>.</mo></mrow></math> It is easy to see that this point is nondegenerate with vanishing T-index, i.e. <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>T</mi><mi>I</mi><mo>=</mo><mn>0</mn></mrow></math> , since <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>a</mi><mn>00</mn></msub><mrow><mo stretchy="false">(</mo><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo stretchy="false">)</mo></mrow><mo>=</mo><mi mathvariant="normal">∅</mi></mrow></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mrow><mi mathvariant="script">T</mi></mrow><mrow><mo stretchy="false">(</mo><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo stretchy="false">)</mo></mrow><mi mathvariant="script">R</mi></msubsup><mo>=</mo><mrow><mo stretchy="false">{</mo><mn>0</mn><mo stretchy="false">}</mo></mrow></mrow></math> . Note that additionally <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mfenced close="}" open="{"><mi>i</mi><mo>∈</mo><msub><mi>a</mi><mn>01</mn></msub><mfenced close=")" open="("><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mfenced><mspace width="0.166667em" /><mfenced open="|"><mspace width="0.166667em" /><msub><mover accent="true"><mrow><mi>σ</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mrow><mn>1</mn><mo>,</mo><mi>i</mi></mrow></msub><mo>=</mo><mn>0</mn></mfenced></mfenced><mo>=</mo><mrow><mo stretchy="false">{</mo><mn>1</mn><mo stretchy="false">}</mo></mrow></mrow></math> . Although all assumptions of Theorem 4 are fulfilled, we have here:

Graph

With other words, the saddle points of the Scholtes-type regularization <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="script">S</mi></math> approximate a minimizer of the regularized continuous reformulation <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="script">R</mi></math> . The reason is that the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>σ</mi></math> -multipliers corresponding to zero x- and nonzero y-variables vanish. The lower bound given in Theorem 4 is attained. <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>□</mo></math>

Next, we point out that the assumption NDT6 is not restrictive at all.

Remark 2

(Genericity for NDT6) Let us briefly sketch why condition NDT6 must be generically fulfilled at the T-stationary points of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="script">R</mi></math> . First, we note that all T-stationary points of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="script">R</mi></math> are generically nondegenerate, see [[9]]. Now, let us count the losses of freedom induced by the definition of a T-stationary point. For feasibility we have <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="|" open="|"><mi>P</mi></mfenced></math> equality constraints, <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="|" open="|"><msub><mi>Q</mi><mn>0</mn></msub></mfenced></math> active inequality constraints, <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="|" open="|"><mi mathvariant="script">E</mi></mfenced></math> bounding constraints on the y-variables, potentially one summation constraint, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mfenced close="|" open="|"><msub><mi>a</mi><mn>01</mn></msub></mfenced><mo>+</mo><mfenced close="|" open="|"><msub><mi>a</mi><mn>10</mn></msub></mfenced><mo>+</mo><mn>2</mn><mfenced close="|" open="|"><msub><mi>a</mi><mn>00</mn></msub></mfenced></mrow></math> orthogonality type constraints. Additional losses of freedom come from the T-stationarity condition. They amount to <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mn>2</mn><mi>n</mi><mo>-</mo><mfenced close="|" open="|"><mi>P</mi></mfenced><mo>-</mo><mfenced close="|" open="|"><msub><mi>Q</mi><mn>0</mn></msub></mfenced><mo>-</mo><mfenced close="|" open="|"><mi mathvariant="script">E</mi></mfenced><mo>-</mo><mn>1</mn><mo>-</mo><mfenced close="|" open="|"><msub><mi>a</mi><mn>01</mn></msub></mfenced><mo>-</mo><mfenced close="|" open="|"><msub><mi>a</mi><mn>10</mn></msub></mfenced><mo>-</mo><mn>2</mn><mfenced close="|" open="|"><msub><mi>a</mi><mn>00</mn></msub></mfenced></mrow></math> if the summation constraint is active, and to <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mn>2</mn><mi>n</mi><mo>-</mo><mfenced close="|" open="|"><mi>P</mi></mfenced><mo>-</mo><mfenced close="|" open="|"><msub><mi>Q</mi><mn>0</mn></msub></mfenced><mo>-</mo><mfenced close="|" open="|"><mi mathvariant="script">E</mi></mfenced><mo>-</mo><mfenced close="|" open="|"><msub><mi>a</mi><mn>01</mn></msub></mfenced><mo>-</mo><mfenced close="|" open="|"><msub><mi>a</mi><mn>10</mn></msub></mfenced><mo>-</mo><mn>2</mn><mfenced close="|" open="|"><msub><mi>a</mi><mn>00</mn></msub></mfenced></mrow></math> otherwise. In both cases, the losses of freedom are equal to the number of variables 2n. The violation of NDT6 would produce an additional loss of freedom, which would imply that the total available degrees of freedom 2n are exceeded. By virtue of the structured jet transversality theorem from [[12]], this cannot happen generically. <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>□</mo></math>

Let us examine the set of multipliers from Theorem 4 in terms of CCOP.

Lemma 4

Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></math> be a nondegenerate M-stationary point of CCOP. Then, for any <math xmlns="http://www.w3.org/1998/Math/MathML"><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo stretchy="false">)</mo></mrow></math> is a T-stationary point of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="script">R</mi></math> we have

Graph

Proof

We refer to the proof of Theorem 3.7 from [[9]]. There, it was shown how any T-stationary point <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo stretchy="false">)</mo></mrow></math> of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="script">R</mi></math> can be constructed by means of a nondegenerate M-stationary point <math xmlns="http://www.w3.org/1998/Math/MathML"><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></math> of CCOP. Specifically, the corresponding multipliers were set as

Graph

We conclude

Graph

Let us assume that <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mfenced close="∥" open="∥"><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mfenced><mn>0</mn></msub><mo><</mo><mi>s</mi></mrow></math> additionally holds. Hence, in virtue of NDM3 we have

Graph

By recalling <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>a</mi><mn>01</mn></msub><mfenced close=")" open="("><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mfenced><mo>⊂</mo><msub><mi>I</mi><mn>0</mn></msub><mfenced close=")" open="("><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mfenced></mrow></math> , the assertion follows.

Suppose now <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mfenced close="∥" open="∥"><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mfenced><mn>0</mn></msub><mo>=</mo><mi>s</mi></mrow></math> instead. Due to Lemma 1b), we have

Graph

Since <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>I</mi><mn>0</mn></msub><mfenced close=")" open="("><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mfenced><mo>=</mo><msub><mi>a</mi><mn>00</mn></msub><mfenced close=")" open="("><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mfenced><mo>∪</mo><msub><mi>a</mi><mn>01</mn></msub><mfenced close=")" open="("><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mfenced></mrow></math> , we conclude <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>a</mi><mn>00</mn></msub><mfenced close=")" open="("><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mfenced><mo>=</mo><mi mathvariant="normal">∅</mi></mrow></math> . Thus, <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>I</mi><mn>0</mn></msub><mfenced close=")" open="("><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mfenced><mo>=</mo><msub><mi>a</mi><mn>01</mn></msub><mfenced close=")" open="("><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mfenced></mrow></math> and the assertion follows. <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>□</mo></math>

In view of Theorem 1 we get the following convergence properties of the proposed Scholtes-type regularization with respect to the underlying CCOP.

Corollary 1

(Convergence from <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="script">S</mi></math> to CCOP)

Suppose that a sequence of Karush–Kuhn–Tucker points <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><msup><mi>x</mi><mi>t</mi></msup><mo>,</mo><msup><mi>y</mi><mi>t</mi></msup><mo stretchy="false">)</mo></mrow></math> of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="script">S</mi></math> converges to <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close=")" open="("><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mfenced></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>t</mi><mo stretchy="false">→</mo><mn>0</mn></mrow></math> . If CC-LICQ holds at <math xmlns="http://www.w3.org/1998/Math/MathML"><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></math> , then <math xmlns="http://www.w3.org/1998/Math/MathML"><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></math> is an M-stationary point of CCOP.

Suppose that a sequence of nondegenerate Karush–Kuhn–Tucker points <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><msup><mi>x</mi><mi>t</mi></msup><mo>,</mo><msup><mi>y</mi><mi>t</mi></msup><mo stretchy="false">)</mo></mrow></math> of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="script">S</mi></math> with quadratic index m converges to <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close=")" open="("><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mfenced></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>t</mi><mo stretchy="false">→</mo><mn>0</mn></mrow></math> . If <math xmlns="http://www.w3.org/1998/Math/MathML"><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></math> is a nondegenerate M-stationary point of CCOP, then we have for its M-index MI:

Graph

If additionally NDM5 holds, then the indices coincide, i.e. <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>M</mi><mi>I</mi><mo>=</mo><mi>m</mi></mrow></math> .

Proof

Due to continuity arguments, <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo stretchy="false">)</mo></mrow></math> is feasible for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="script">R</mi></math> . Let us show that the latter implies feasibility of <math xmlns="http://www.w3.org/1998/Math/MathML"><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></math> for CCOP. For this purpose we assume instead <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mfenced close="∥" open="∥"><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mfenced><mn>0</mn></msub><mo>></mo><mi>s</mi></mrow></math> . Consequently, we have

Graph

Thus, it holds for <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo stretchy="false">)</mo></mrow></math> :

Graph

a contradiction to its feasibility. Overall, <math xmlns="http://www.w3.org/1998/Math/MathML"><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></math> has to be feasible for CCOP and, thus, we can apply Proposition 3.2a) from [[9]]. The latter states that if <math xmlns="http://www.w3.org/1998/Math/MathML"><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></math> is feasible for CCOP and satisfies CC-LICQ, then MPOC-LICQ holds at any <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></math> that is feasible for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="script">R</mi></math> . Hence, in view of Theorem 3, <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo stretchy="false">)</mo></mrow></math> is a T-stationary point of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="script">R</mi></math> . Therefore, <math xmlns="http://www.w3.org/1998/Math/MathML"><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></math> is M-stationary, due to Theorem 1b).

We deduce as above that <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo stretchy="false">)</mo></mrow></math> is a T-stationary point of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="script">R</mi></math> . Using Theorem 1, we have that <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo stretchy="false">)</mo></mrow></math> is nondegenerate fulfilling NDT5. According to Theorem 4, for its T-index TI it holds

Graph

However, we again use Theorem 1 to conclude <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>T</mi><mi>I</mi><mo>=</mo><mi>M</mi><mi>I</mi></mrow></math> . In view of Lemma 4, the assertion follows.

Let us briefly comment on condition NDM5. It ensures M-stationary points to have the same index as the approximating Karush–Kuhn–Tucker points of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="script">S</mi></math> .

Remark 3

(On condition NDM5) It follows from Lemma 4 that for a nondegenerate M-stationary point <math xmlns="http://www.w3.org/1998/Math/MathML"><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></math> of CCOP the following statements are equivalent:

NDM5 holds at <math xmlns="http://www.w3.org/1998/Math/MathML"><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></math> ,

NDT6 holds at a T-stationary point <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo stretchy="false">)</mo></mrow></math> of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="script">R</mi></math> ,

NDT6 holds at all T-stationary points <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo stretchy="false">)</mo></mrow></math> of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="script">R</mi></math> .

Further, due to Theorem 1a), all M-stationary points are induced by at least one T-stationary point. Theorem 1b), Remark 2, and the equivalence above provide that all M-stationary points generically fulfill NDM5. As a consequence, we conclude that generically the bounds given in Corollary 1 are tight, i.e. <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>M</mi><mi>I</mi><mo>=</mo><mi>m</mi></mrow></math> . The latter holds in particular for <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mo stretchy="false">‖</mo><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo stretchy="false">‖</mo></mrow><mn>0</mn></msub><mo><</mo><mi>s</mi></mrow></math> regardless of NDM5, since NDM3 suffices. <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>□</mo></math>

Now, we prove that the Scholtes-type regularization method is well-defined. For the proof see again the Appendix below.

Theorem 5

(Well-posedness of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="script">S</mi></math> from <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="script">R</mi></math> ) Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo stretchy="false">)</mo></mrow></math> be a nondegenerate T-stationary point of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="script">R</mi></math> with T-index m, additionally, fulfilling NDT6. Then, for all sufficiently small t there exists a nondegenerate Karush–Kuhn–Tucker point <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><msup><mi>x</mi><mi>t</mi></msup><mo>,</mo><msup><mi>y</mi><mi>t</mi></msup><mo stretchy="false">)</mo></mrow></math> of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="script">S</mi></math> within a neighborhood of <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo stretchy="false">)</mo></mrow></math> , which has the same quadratic index m. Moreover, for any fixed t sufficiently small, such <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><msup><mi>x</mi><mi>t</mi></msup><mo>,</mo><msup><mi>y</mi><mi>t</mi></msup><mo stretchy="false">)</mo></mrow></math> is the unique Karush–Kuhn–Tucker point of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="script">S</mi></math> in a sufficiently small neighborhood of <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo stretchy="false">)</mo></mrow></math> .

Again, we state the result analogous to Theorem 5 in terms of CCOP.

Corollary 2

(Well-posedness of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="script">S</mi></math> from CCOP) Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></math> be a nondegenerate M-stationary point of CCOP with M-index m, additionally, fulfilling NDM5. Then, for all sufficiently small t there exists a nondegenerate Karush–Kuhn–Tucker point <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><msup><mi>x</mi><mi>t</mi></msup><mo>,</mo><msup><mi>y</mi><mi>t</mi></msup><mo stretchy="false">)</mo></mrow></math> of S with <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mi>t</mi></msup></math> being within a neighborhood of <math xmlns="http://www.w3.org/1998/Math/MathML"><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></math> , which has the same quadratic index m.

Proof

Due to Theorem 1a), there exists at least one nondegenerate T-stationary point <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo stretchy="false">)</mo></mrow></math> of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="script">R</mi></math> . Moreover, Lemma 4 provides that it also fulfills NDT6. Thus, the assertion follows straightforward in view of Theorem 5. <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>□</mo></math>

Let us compare our results with those for the initially proposed continuous reformulation (1) and the Scholtes-type relaxation (2) from [[5]] and [[6]], respectively. There, the concept of S-stationarity for (1) becomes crucial.

Definition 13

(S-stationary, [[5]]) A feasible point <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo stretchy="false">)</mo></mrow></math> of (1) is called S-stationary if there exist multipliers

Graph

such that the following conditions hold:

Graph

Example 3

We consider the following CCOP with <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>n</mi><mo>=</mo><mn>3</mn></mrow></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>s</mi><mo>=</mo><mn>1</mn></mrow></math> :

Graph

It has minimizers at (1, 0, 0), (0, 1, 0), and (0, 0, 1) as well as a saddle point at (0, 0, 0). It is straightforward to check that all these points are nondegenerate M-stationary points, which additionally fulfill NDM5. For its continuous reformulation (1) we have

Graph

We get as its S-stationary points:

Graph

and

Graph

Hence, we have a continuum of saddle points. Moreover, it was shown in [[4]] that all S-stationary points of reformulation (1) are degenerate T-stationary points, i.e. violating at least one of the conditions NDT1-NDT4. Further, we turn our attention to the Scholtes-type regularization (2)

Graph

For t sufficiently small its Karush–Kuhn–Tucker points include

Graph

where

Graph

For every minimizer of the underlying CCOP there exist at least two sequences of Karush–Kuhn–Tucker points of the Scholtes-type regularization (2), which approximate the corresponding S-stationary points of the continuous reformulation (1), i.e.

Graph

For the saddle point of CCOP there exist at least four sequences of Karush–Kuhn–Tucker points of the Scholtes-type regularization (2), which approximate the corresponding S-stationary points of the continuous reformulation (1), i.e.

Graph

We list M-, S-stationary, and Karush–Kuhn–Tucker points in Table 1.

Table 1 M-, S-stationary, and Karush–Kuhn–Tucker points

<table frame="hsides" rules="groups"><thead><tr><th align="left">M-stationary for CCOP</th><th align="left">S-stationary for (1)</th><th align="left">Karush–Kuhn–Tucker points for (2)</th></tr></thead><tbody><tr><td align="left">(1, 0, 0)</td><td align="left">(1, 0, 0, 0, 1, 1)</td><td align="left"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup xmlns=""><mi>z</mi><mrow><mn>1</mn><mo>,</mo><mi>t</mi></mrow></msup></math><inline-graphic mime-subtype="GIF" href="10107_2024_2082_Article_IEq413.gif" />, <math xmlns="http://www.w3.org/1998/Math/MathML"><msup xmlns=""><mi>z</mi><mrow><mn>2</mn><mo>,</mo><mi>t</mi></mrow></msup></math><inline-graphic mime-subtype="GIF" href="10107_2024_2082_Article_IEq414.gif" /></td></tr><tr><td align="left">(0, 1, 0)</td><td align="left">(0, 1, 0, 1, 0, 1)</td><td align="left"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup xmlns=""><mi>z</mi><mrow><mn>3</mn><mo>,</mo><mi>t</mi></mrow></msup></math><inline-graphic mime-subtype="GIF" href="10107_2024_2082_Article_IEq415.gif" />, <math xmlns="http://www.w3.org/1998/Math/MathML"><msup xmlns=""><mi>z</mi><mrow><mn>4</mn><mo>,</mo><mi>t</mi></mrow></msup></math><inline-graphic mime-subtype="GIF" href="10107_2024_2082_Article_IEq416.gif" /></td></tr><tr><td align="left">(0, 0, 1)</td><td align="left">(0, 0, 1, 1, 1, 0)</td><td align="left"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup xmlns=""><mi>z</mi><mrow><mn>5</mn><mo>,</mo><mi>t</mi></mrow></msup></math><inline-graphic mime-subtype="GIF" href="10107_2024_2082_Article_IEq417.gif" />, <math xmlns="http://www.w3.org/1998/Math/MathML"><msup xmlns=""><mi>z</mi><mrow><mn>6</mn><mo>,</mo><mi>t</mi></mrow></msup></math><inline-graphic mime-subtype="GIF" href="10107_2024_2082_Article_IEq418.gif" /></td></tr><tr><td align="left" rowspan="4">(0, 0, 0)</td><td align="left"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mfrac bevelled="true"><mn>1</mn><mn>2</mn></mfrac><mo>,</mo><mfrac bevelled="true"><mn>1</mn><mn>2</mn></mfrac><mo stretchy="false">)</mo></mrow></math><inline-graphic mime-subtype="GIF" href="10107_2024_2082_Article_IEq419.gif" /></td><td align="left"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup xmlns=""><mi>z</mi><mrow><mn>7</mn><mo>,</mo><mi>t</mi></mrow></msup></math><inline-graphic mime-subtype="GIF" href="10107_2024_2082_Article_IEq420.gif" /></td></tr><tr><td align="left"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo>,</mo><mfrac bevelled="true"><mn>1</mn><mn>2</mn></mfrac><mo>,</mo><mn>1</mn><mo>,</mo><mfrac bevelled="true"><mn>1</mn><mn>2</mn></mfrac><mo stretchy="false">)</mo></mrow></math><inline-graphic mime-subtype="GIF" href="10107_2024_2082_Article_IEq421.gif" /></td><td align="left"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup xmlns=""><mi>z</mi><mrow><mn>8</mn><mo>,</mo><mi>t</mi></mrow></msup></math><inline-graphic mime-subtype="GIF" href="10107_2024_2082_Article_IEq422.gif" /></td></tr><tr><td align="left"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo>,</mo><mfrac bevelled="true"><mn>1</mn><mn>2</mn></mfrac><mo>,</mo><mfrac bevelled="true"><mn>1</mn><mn>2</mn></mfrac><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></math><inline-graphic mime-subtype="GIF" href="10107_2024_2082_Article_IEq423.gif" /></td><td align="left"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup xmlns=""><mi>z</mi><mrow><mn>9</mn><mo>,</mo><mi>t</mi></mrow></msup></math><inline-graphic mime-subtype="GIF" href="10107_2024_2082_Article_IEq424.gif" /></td></tr><tr><td align="left"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo>,</mo><mfrac bevelled="true"><mn>2</mn><mn>3</mn></mfrac><mo>,</mo><mfrac bevelled="true"><mn>2</mn><mn>3</mn></mfrac><mo>,</mo><mfrac bevelled="true"><mn>2</mn><mn>3</mn></mfrac><mo stretchy="false">)</mo></mrow></math><inline-graphic mime-subtype="GIF" href="10107_2024_2082_Article_IEq425.gif" /></td><td align="left"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup xmlns=""><mi>z</mi><mrow><mn>10</mn><mo>,</mo><mi>t</mi></mrow></msup></math><inline-graphic mime-subtype="GIF" href="10107_2024_2082_Article_IEq426.gif" /></td></tr></tbody></table>

Let us apply our results to the given CCOP. In view of Theorem 1a), we know that the regularized continuous reformulation <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="script">R</mi></math> has in total five T-stationary points, all of them being nondegenerate. Three of them are minimizers and two of them are saddle points of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="script">R</mi></math> . Also, we know from Remark 3 that all of them fulfill NDT6. Due to Theorem 3, any convergent sequence of Karush–Kuhn–Tucker points of the Scholtes-type regularization <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="script">S</mi></math> converges to one of these T-stationary points of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="script">R</mi></math> . We apply Theorem 5 to conclude that for any fixed t sufficiently small there are exactly five Karush–Kuhn–Tucker points of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="script">S</mi></math> . All of them are nondegenerate. Three of them are minimizers and two of them are saddle points of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="script">S</mi></math> . Overall, not only the global structure of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="script">R</mi></math> is more accessible than that of (1), but also the global structure of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="script">S</mi></math> is more accessible than that of (2). This shows the advantage of our approach in comparison to the existing literature, at least for the presented example. <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>□</mo></math>

Conclusions

In [[9]], the number of saddle points for the regularized continuous reformulation of CCOP has been estimated. Namely, each saddle point of CCOP generates exponentially many saddle points of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="script">R</mi></math> , all of them having the same index. It has been concluded there that the introduction of auxiliary y-variables shifts the complexity of dealing with the cardinality constraint in CCOP into the appearance of multiple saddle points for its continuous reformulation. From our extended convergence analysis of the Scholtes-type regularization it follows that the number of its saddle points also grows exponentially as compared to that of CCOP. We emphasize that this issue is at the core of numerical difficulties if solving CCOP up to global optimality by means of the Scholtes-type regularization method. To the best of our knowledge this is the first paper studying convergence properties of the Scholtes-type regularization method in the vicinity of saddle points, rather than of minimizers. The ideas from our analysis can be potentially applied not only for classes of nonsmooth optimization problems, such as MPCC, MPVC, MPSC, and MPOC, but also for other regularization schemes known from the literature.

Acknowledgements

The authors would like to thank the anonymous referees for suggesting valuable improvements of the paper.

Funding

Open Access funding enabled and organized by Projekt DEAL.

Declarations

Conflict of interest

The authors have no relevant financial or non-financial interests to disclose.

Appendix

Proof of Theorem 4

The proof will be divided into 4 major steps.

<bold> Step 1a. </bold> We rewrite the tangent space corresponding to the Karush–Kuhn–Tucker point

Graph

. For that, we use Lemma 2a) which provides that the summation constraint is active:

Graph

In total there are, due to LICQ,

Graph

linearly independent vectors involved. We use Lemma 3a) and 3b) to substitute

Graph

• with

Graph

• and

Graph

• with

Graph

, respectively. The latter set has cardinality of

Graph

due to Lemma 1c). Additionally, we use Lemma 3c) and 3d) to conclude:

Graph

Finally,

Graph

, cf. Lemma 1b). Thus, we have:

Graph

<bold> Step 1b. </bold> We examine the tangent space corresponding to the T-stationary point

Graph

. For this purpose, we consider the following vectors from its definition:

Graph

The latter vector is involved due to Lemma 1a). The number of these vectors is due to Lemma 1c) equal to

Graph

. Moreover, they are linearly independent due to MPOC-LICQ. Hence, we can write the respective tangent space as

Graph

In total there are, due to MPOC-LICQ,

Graph

linearly independent vectors involved.

<bold> Step 2. </bold> Let

Graph

be a linear subspace. We denote the number of negative eigenvalues of

Graph

• by

Graph

. Analogously,

Graph

stands for the number of negative eigenvalues of

Graph

• and

Graph

stands for the number of negative eigenvalues of

Graph

. We have the following relation between the involved Hessians of the Lagrange functions by denoting

Graph

• ,

Graph

• : 14

Graph

<bold> Step 2a. </bold> It holds for t sufficiently small:

Graph

Indeed, by using (14), we derive for any

Graph

• : 15

Graph

• since

Graph

as seen in Step 1b. Hence, we get

Graph

. Due to NDT4, continuity arguments provide

Graph

• .

<bold> Step 2b. </bold> We claim that the numbers of positive and negative eigenvalues of

Graph

• and of

Graph

, respectively, coincide, where

Graph

• Let

Graph

be the positive eigenvalues of

Graph

with corresponding eigenvectors

Graph

. Hence, for all

Graph

• :

Graph

We rewrite the tangent space

Graph

Due to MPOC-LICQ, the application of the implicit function theorem provides the existence of

Graph

such that for all

Graph

• and

Graph

there exists

Graph

• with

Graph

• and

Graph

. We can choose t even smaller, such that

Graph

remain linearly independent and for all

Graph

it holds:

Graph

• Hence,

Graph

has at least

Graph

positive eigenvalues. If we repeat the above reasoning for negative eigenvalues, the matrix

Graph

has at least as many negative eigenvalues as

Graph

. Additionally, we show that the dimensions of

Graph

• and

Graph

coincide. By Step 1b, we have

Graph

for the dimension of

Graph

. Since MPOC-LICQ remains valid in the neighborhood of

Graph

, we get again

Graph

for the dimension of

Graph

. By continuity arguments, NDT4 and (15) provide that

Graph

is nonsingular. Altogether, the assertion follows.

<bold> Step 2c. </bold> We claim that

Graph

For that, we focus on the dimension of

Graph

. As a consequence of Step 1a it is

Graph

. Due to continuity arguments, we can choose t small enough to ensure

Graph

• ,

Graph

• and

Graph

• ,

Graph

. Using this and Lemma 3a), 3b), and 3d), it follows that

Graph

. Therefore, using Step 2b,

Graph

. We observe in view of NDT4 and Step 1b that

Graph

. The assertion follows immediately.

<bold> Step 3. </bold> Let us show that

Graph

In view of Step 2a, Step 2c, and due to continuity, we have for t sufficiently small:

Graph

We show for t sufficiently small:

Graph

and the assertion will follow immediately since

Graph

. Clearly,

Graph

• Suppose

Graph

• with

Graph

. In view of Remark 1, the difference

Graph

cannot vanish for all t sufficiently small. In particular, one of the multipliers

Graph

• or

Graph

has to be not vanishing for all t sufficiently small. Hence,

Graph

. We therefore have:

Graph

<bold> Step 4. </bold> Without loss of generality—considering subsequences if needed—we can assume that for any

Graph

at least one of the sequences

Graph

• or

Graph

is convergent. First, we note that the quotients are well defined due to Lemma 3c). Moreover, if the former sequence does not contain a convergent subsequence, we find a subsequence that tends to plus or minus infinity. Consequently, the corresponding subsequence of the latter reciprocal sequence has to converge to zero. We define the following auxiliary sets:

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtable><mtr><mtd columnalign="right"><mrow><msubsup><mi>a</mi><mrow><mn>00</mn></mrow><mi>x</mi></msubsup><mrow><mo stretchy="false">(</mo><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo stretchy="false">)</mo></mrow><mo>=</mo></mrow></mtd><mtd columnalign="left"><mrow><mrow /><mfenced close="}" open="{"><mi>i</mi><mo>∈</mo><msub><mi>a</mi><mn>00</mn></msub><mrow><mo stretchy="false">(</mo><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo stretchy="false">)</mo></mrow><mspace width="0.166667em" /><mfenced open="|"><mspace width="0.166667em" /><mfrac><msubsup><mi>x</mi><mi>i</mi><mi>t</mi></msubsup><msubsup><mi>y</mi><mi>i</mi><mi>t</mi></msubsup></mfrac><mspace width="0.333333em" /><mtext>converges</mtext><mspace width="0.333333em" /><mspace width="0.333333em" /><mtext>for</mtext><mspace width="0.333333em" /><mi>t</mi><mo stretchy="false">→</mo><mn>0</mn></mfenced></mfenced><mo>,</mo></mrow></mtd></mtr><mtr><mtd columnalign="right"><mrow><mrow /><msubsup><mi>a</mi><mrow><mn>00</mn></mrow><mi>y</mi></msubsup><mrow><mo stretchy="false">(</mo><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo stretchy="false">)</mo></mrow><mo>=</mo></mrow></mtd><mtd columnalign="left"><mrow><mrow /><msub><mi>a</mi><mn>00</mn></msub><mrow><mrow><mo stretchy="false">(</mo><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo stretchy="false">)</mo></mrow><mo stretchy="true">\</mo></mrow><msubsup><mi>a</mi><mrow><mn>00</mn></mrow><mi>x</mi></msubsup><mrow><mo stretchy="false">(</mo><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo stretchy="false">)</mo></mrow><mo>.</mo></mrow></mtd></mtr></mtable></mrow></math>

Graph

• For

Graph

we consider

Graph

and replace two of the involved equations, namely

Graph

• and

Graph

by one equation

Graph

. Clearly, the vectors involved in the definition of the newly generated linear space, i.e.

Graph

remain linearly independent. The dimension of

Graph

is greater than the dimension of

Graph

by one. Moreover, there exists

Graph

• with

Graph

. Indeed, assume that no such

Graph

exists, then we can add the equation

Graph

to the defining equations of

Graph

without changing it. The resulting space, however, is identical to

Graph

, a contradiction. Without loss of generality, we assume

Graph

. Further, by straightforward application of the implicit function theorem and due to Lemma 3a) and 3b), we find a sequence of vectors

Graph

that converges to

Graph

• for

Graph

. For this, we define

Graph

We again have, due to continuity arguments, that

Graph

• . For

Graph

we proceed analogously by considering

Graph

again and replace two of the involved equations

Graph

• and

Graph

by the equation

Graph

. By the same arguments as before, we find

Graph

• with

Graph

. Again we will assume

Graph

and find a sequence of vectors

Graph

that converges to

Graph

• for

Graph

. Due to continuity, it holds then

Graph

• .

It is straightforward to verify the following observations for t sufficiently small:

Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="}" open="{"><msup><mi>ξ</mi><mrow><mo>′</mo><mo>,</mo><mn>1</mn></mrow></msup><mo>,</mo><mo>...</mo><mo>,</mo><msup><mi>ξ</mi><mrow><mo>′</mo><mo>,</mo><mi>ℓ</mi></mrow></msup></mfenced></math>

Graph

be a base of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mrow><mi mathvariant="script">T</mi></mrow><mo>′</mo></msup></math>

Graph

, cf. Step 2b, then <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mfenced close="}" open="{"><msup><mi>ξ</mi><mrow><mo>′</mo><mo>,</mo><mn>1</mn></mrow></msup><mo>,</mo><mo>...</mo><mo>,</mo><msup><mi>ξ</mi><mrow><mo>′</mo><mo>,</mo><mi>ℓ</mi></mrow></msup></mfenced><mo>∪</mo><mfenced close="}" open="{"><msubsup><mi>ξ</mi><mi>t</mi><mover accent="true"><mrow><mi>i</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></msubsup><mspace width="0.166667em" /><mfenced open="|"><mspace width="0.166667em" /><mover accent="true"><mrow><mi>i</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>∈</mo><msub><mi>a</mi><mn>00</mn></msub><mrow><mo stretchy="false">(</mo><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo stretchy="false">)</mo></mrow></mfenced></mfenced></mrow></math>

Graph

is a set of linear independent vectors. In fact, suppose for some coefficients <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>b</mi><mi>i</mi></msub><mo>∈</mo><mi mathvariant="double-struck">R</mi></mrow></math>

Graph

, <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>i</mi><mo>∈</mo><msub><mi>a</mi><mn>00</mn></msub><mfenced close=")" open="("><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mfenced><mo>,</mo><msub><mi>β</mi><mi>i</mi></msub><mo>∈</mo><mi mathvariant="double-struck">R</mi><mo>,</mo><mspace width="0.166667em" /><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo>...</mo><mo>,</mo><mi>ℓ</mi></mrow></math>

Graph

it holds:

Graph

For <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mover accent="true"><mrow><mi>i</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>∈</mo><msubsup><mi>a</mi><mrow><mn>00</mn></mrow><mi>x</mi></msubsup><mfenced close=")" open="("><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mfenced></mrow></math>

Graph

we consider the <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mover accent="true"><mrow><mi>i</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo stretchy="false">)</mo></mrow></math>

Graph

-th row of this sum

Graph

If instead <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mover accent="true"><mrow><mi>i</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>∈</mo><msubsup><mi>a</mi><mrow><mn>00</mn></mrow><mi>y</mi></msubsup><mfenced close=")" open="("><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mfenced></mrow></math>

Graph

we consider the <math xmlns="http://www.w3.org/1998/Math/MathML"><mover accent="true"><mrow><mi>i</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></math>

Graph

-th row of the sum

Graph

Altogether, it must hold <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>b</mi><mi>i</mi></msub><mo>=</mo><mn>0</mn></mrow></math>

Graph

. However, this implies

Graph

Hence, <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>β</mi><mi>i</mi></msub><mo>=</mo><mn>0</mn></mrow></math>

Graph

for <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo>...</mo><mo>,</mo><mi>ℓ</mi></mrow></math>

Graph

It holds <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mi>ξ</mi><mi>t</mi><mover accent="true"><mrow><mi>i</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></msubsup><mo>∈</mo><msubsup><mrow><mi mathvariant="script">T</mi></mrow><mfenced close=")" open="("><msup><mi>x</mi><mi>t</mi></msup><mo>,</mo><msup><mi>y</mi><mi>t</mi></msup></mfenced><mi mathvariant="script">S</mi></msubsup></mrow></math>

Graph

, cf. Step 1a, for any <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mover accent="true"><mrow><mi>i</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>∈</mo><msub><mi>a</mi><mn>00</mn></msub><mrow><mo stretchy="false">(</mo><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo stretchy="false">)</mo></mrow></mrow></math>

Graph

It holds <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mfenced close=")" open="("><msubsup><mi>η</mi><mi>i</mi><mrow><mo>≤</mo><mo>,</mo><mi>t</mi></mrow></msubsup><mo>-</mo><msubsup><mi>η</mi><mi>i</mi><mrow><mo>≥</mo><mo>,</mo><mi>t</mi></mrow></msubsup></mfenced><msubsup><mi>ξ</mi><mrow><mi>t</mi><mo>,</mo><mi>i</mi></mrow><mover accent="true"><mrow><mi>i</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></msubsup><msubsup><mi>ξ</mi><mrow><mi>t</mi><mo>,</mo><mi>n</mi><mo>+</mo><mi>i</mi></mrow><mover accent="true"><mrow><mi>i</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></msubsup><mo>≤</mo><mn>0</mn><mo>,</mo><mi>i</mi><mo>∈</mo><mi mathvariant="script">H</mi><mfenced close=")" open="("><msup><mi>x</mi><mi>t</mi></msup><mo>,</mo><msup><mi>y</mi><mi>t</mi></msup></mfenced></mrow></math>

Graph

. Since <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mi>ξ</mi><mi>t</mi><mover accent="true"><mrow><mi>i</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></msubsup><mo>∈</mo><msubsup><mrow><mi mathvariant="script">T</mi></mrow><mfenced close=")" open="("><msup><mi>x</mi><mi>t</mi></msup><mo>,</mo><msup><mi>y</mi><mi>t</mi></msup></mfenced><mi mathvariant="script">S</mi></msubsup></mrow></math>

Graph

, we obtain:

Graph

If <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>i</mi><mo>∈</mo><msup><mrow><mi mathvariant="script">H</mi></mrow><mo>≥</mo></msup><mfenced close=")" open="("><msup><mi>x</mi><mi>t</mi></msup><mo>,</mo><msup><mi>y</mi><mi>t</mi></msup></mfenced></mrow></math>

Graph

, we have <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mi>x</mi><mi>i</mi><mi>t</mi></msubsup><mo><</mo><mn>0</mn><mo>,</mo><msubsup><mi>y</mi><mi>i</mi><mi>t</mi></msubsup><mo>></mo><mn>0</mn></mrow></math>

Graph

and <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mi>η</mi><mi>i</mi><mrow><mo>≤</mo><mo>,</mo><mi>t</mi></mrow></msubsup><mo>=</mo><mn>0</mn></mrow></math>

Graph

. Moreover, due to ND2, we have <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mi>η</mi><mi>i</mi><mrow><mo>≥</mo><mo>,</mo><mi>t</mi></mrow></msubsup><mo>></mo><mn>0</mn></mrow></math>

Graph

. The assertion follows immediately. The other case <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>i</mi><mo>∈</mo><msup><mrow><mi mathvariant="script">H</mi></mrow><mo>≤</mo></msup><mfenced close=")" open="("><msup><mi>x</mi><mi>t</mi></msup><mo>,</mo><msup><mi>y</mi><mi>t</mi></msup></mfenced></mrow></math>

Graph

is completely analogous.

It holds <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><munder><mo movablelimits="false">lim</mo><mrow><mi>t</mi><mo stretchy="false">→</mo><mn>0</mn></mrow></munder><mfenced close=")" open="("><msubsup><mi>η</mi><mrow><mover accent="true"><mrow><mi>i</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mrow><mrow><mo>≤</mo><mo>,</mo><mi>t</mi></mrow></msubsup><mo>-</mo><msubsup><mi>η</mi><mrow><mover accent="true"><mrow><mi>i</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mrow><mrow><mo>≥</mo><mo>,</mo><mi>t</mi></mrow></msubsup></mfenced><msubsup><mi>ξ</mi><mrow><mi>t</mi><mo>,</mo><mover accent="true"><mrow><mi>i</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mrow><mover accent="true"><mrow><mi>i</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></msubsup><msubsup><mi>ξ</mi><mrow><mi>t</mi><mo>,</mo><mi>n</mi><mo>+</mo><mover accent="true"><mrow><mi>i</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mrow><mover accent="true"><mrow><mi>i</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></msubsup><mo>=</mo><mo>-</mo><mi>∞</mi></mrow></math>

Graph

. We calculate:

Graph

Let us suppose <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mover accent="true"><mrow><mi>i</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>∈</mo><msubsup><mi>a</mi><mrow><mn>00</mn></mrow><mi>x</mi></msubsup><mrow><mo stretchy="false">(</mo><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo stretchy="false">)</mo></mrow></mrow></math>

Graph

. We have <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mi>y</mi><mover accent="true"><mrow><mi>i</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mi>t</mi></msubsup><mo>≠</mo><mn>0</mn></mrow></math>

Graph

and, thus, <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mi>ν</mi><mover accent="true"><mrow><mi>i</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mi>t</mi></msubsup><mo>=</mo><mn>0</mn></mrow></math>

Graph

. We use Remark 1 and NDT3 to conclude that the sequence <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mfenced close=")" open="("><msubsup><mi>η</mi><mrow><mover accent="true"><mrow><mi>i</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mrow><mrow><mo>≥</mo><mo>,</mo><mi>t</mi></mrow></msubsup><mo>-</mo><msubsup><mi>η</mi><mrow><mover accent="true"><mrow><mi>i</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mrow><mrow><mo>≤</mo><mo>,</mo><mi>t</mi></mrow></msubsup></mfenced><msubsup><mi>x</mi><mover accent="true"><mrow><mi>i</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mi>t</mi></msubsup></mrow></math>

Graph

converges to <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>ϱ</mi><mrow><mn>2</mn><mo>,</mo><mover accent="true"><mrow><mi>i</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mrow></msub><mo><</mo><mn>0</mn></mrow></math>

Graph

for <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>t</mi><mo stretchy="false">→</mo><mn>0</mn></mrow></math>

Graph

. Further, <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mfrac><mn>1</mn><msubsup><mi>y</mi><mover accent="true"><mrow><mi>i</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mi>t</mi></msubsup></mfrac><mo>></mo><mn>0</mn></mrow></math>

Graph

tends to infinity for <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>t</mi><mo stretchy="false">→</mo><mn>0</mn></mrow></math>

Graph

. Finally, <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced close=")" open="("><msubsup><mi>ξ</mi><mrow><mi>t</mi><mo>,</mo><mi>n</mi><mo>+</mo><mover accent="true"><mrow><mi>i</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mrow><mover accent="true"><mrow><mi>i</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></msubsup></mfenced><mn>2</mn></msup></math>

Graph

converges to 1 for <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>t</mi><mo stretchy="false">→</mo><mn>0</mn></mrow></math>

Graph

, due to the construction of <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>ξ</mi><mi>t</mi><mover accent="true"><mrow><mi>i</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></msubsup></math>

Graph

. Thus, the assertion follows. Instead, let us suppose <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mover accent="true"><mrow><mi>i</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>∈</mo><msubsup><mi>a</mi><mrow><mn>00</mn></mrow><mi>y</mi></msubsup><mrow><mo stretchy="false">(</mo><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo stretchy="false">)</mo></mrow></mrow></math>

Graph

. This time, we have that <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced close=")" open="("><msubsup><mi>ξ</mi><mrow><mi>t</mi><mo>,</mo><mover accent="true"><mrow><mi>i</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mrow><mover accent="true"><mrow><mi>i</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></msubsup></mfenced><mn>2</mn></msup></math>

Graph

converges to 1 for <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>t</mi><mo stretchy="false">→</mo><mn>0</mn></mrow></math>

Graph

. Due to Remark 1 and NDT3, <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mfenced close=")" open="("><msubsup><mi>η</mi><mrow><mover accent="true"><mrow><mi>i</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mrow><mrow><mo>≥</mo><mo>,</mo><mi>t</mi></mrow></msubsup><mo>-</mo><msubsup><mi>η</mi><mrow><mover accent="true"><mrow><mi>i</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mrow><mrow><mo>≤</mo><mo>,</mo><mi>t</mi></mrow></msubsup></mfenced><msubsup><mi>y</mi><mover accent="true"><mrow><mi>i</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mi>t</mi></msubsup></mrow></math>

Graph

converges to <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>ϱ</mi><mrow><mn>1</mn><mo>,</mo><mover accent="true"><mrow><mi>i</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mrow></msub><mo>≠</mo><mn>0</mn></mrow></math>

Graph

for <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>t</mi><mo stretchy="false">→</mo><mn>0</mn></mrow></math>

Graph

. If <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mover accent="true"><mrow><mi>i</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>∈</mo><msup><mrow><mi mathvariant="script">H</mi></mrow><mo>≥</mo></msup><mfenced close=")" open="("><msup><mi>x</mi><mi>t</mi></msup><mo>,</mo><msup><mi>y</mi><mi>t</mi></msup></mfenced></mrow></math>

Graph

, then <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ϱ</mi><mrow><mn>1</mn><mo>,</mo><mover accent="true"><mrow><mi>i</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mrow></msub></math>

Graph

is positive from here. Also, <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mfrac><mn>1</mn><msubsup><mi>x</mi><mover accent="true"><mrow><mi>i</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mi>t</mi></msubsup></mfrac><mo><</mo><mn>0</mn></mrow></math>

Graph

tends to minus infinity for <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>t</mi><mo stretchy="false">→</mo><mn>0</mn></mrow></math>

Graph

. The other case <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mover accent="true"><mrow><mi>i</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>∈</mo><msup><mrow><mi mathvariant="script">H</mi></mrow><mo>≤</mo></msup><mfenced close=")" open="("><msup><mi>x</mi><mi>t</mi></msup><mo>,</mo><msup><mi>y</mi><mi>t</mi></msup></mfenced></mrow></math>

Graph

is completely analogous.

We notice that <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="true" scriptlevel="0"><mrow><msubsup><mi>ξ</mi><mi>t</mi><msup><mrow><mover accent="true"><mrow><mi>i</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mrow><mi>T</mi></msup></msubsup><msup><mi>D</mi><mn>2</mn></msup><mspace width="0.166667em" /><msup><mi>L</mi><mi mathvariant="script">R</mi></msup><mfenced close=")" open="("><msup><mi>x</mi><mi>t</mi></msup><mo>,</mo><msup><mi>y</mi><mi>t</mi></msup></mfenced><msubsup><mi>ξ</mi><mi>t</mi><mover accent="true"><mrow><mi>i</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></msubsup></mrow></mstyle></math>

Graph

converges for <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>t</mi><mo stretchy="false">→</mo><mn>0</mn></mrow></math>

Graph

due to the construction above.

Finally, for <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mover accent="true"><mrow><mi>i</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>∈</mo><msub><mi>a</mi><mn>00</mn></msub><mrow><mo stretchy="false">(</mo><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo stretchy="false">)</mo></mrow></mrow></math>

Graph

we estimate:

Graph

Thus, due to d) and e), <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mi>ξ</mi><mi>t</mi><msup><mrow><mover accent="true"><mrow><mi>i</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mrow><mi>T</mi></msup></msubsup><msup><mi>D</mi><mn>2</mn></msup><mspace width="0.166667em" /><msup><mi>L</mi><mi mathvariant="script">S</mi></msup><mfenced close=")" open="("><msup><mi>x</mi><mi>t</mi></msup><mo>,</mo><msup><mi>y</mi><mi>t</mi></msup></mfenced><msubsup><mi>ξ</mi><mi>t</mi><mover accent="true"><mrow><mi>i</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></msubsup></mrow></math>

Graph

has to be negative for t small enough. As we have seen in Step 2c, it holds <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mrow><mi mathvariant="script">T</mi></mrow><mo>′</mo></msup><mo>⊂</mo><msubsup><mrow><mi mathvariant="script">T</mi></mrow><mfenced close=")" open="("><msup><mi>x</mi><mi>t</mi></msup><mo>,</mo><msup><mi>y</mi><mi>t</mi></msup></mfenced><mi mathvariant="script">S</mi></msubsup></mrow></math>

Graph

. Then, due a) and b), we have

Graph

By Step 2a, we have <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>Q</mi><msubsup><mi>I</mi><mrow><mi>t</mi><mo>,</mo><msubsup><mrow><mi mathvariant="script">T</mi></mrow><mfenced close=")" open="("><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mfenced><mi mathvariant="script">R</mi></msubsup></mrow><mi mathvariant="script">S</mi></msubsup><mo>=</mo><msubsup><mover><mrow><mi mathvariant="italic">QI</mi></mrow><mo>¯</mo></mover><msubsup><mrow><mi mathvariant="script">T</mi></mrow><mfenced close=")" open="("><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mfenced><mi mathvariant="script">R</mi></msubsup><mi mathvariant="script">R</mi></msubsup></mrow></math>

Graph

, and by Step 2b, <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>Q</mi><msubsup><mi>I</mi><mrow><mi>t</mi><mo>,</mo><msubsup><mrow><mi mathvariant="script">T</mi></mrow><mfenced close=")" open="("><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mfenced><mi mathvariant="script">R</mi></msubsup></mrow><mi mathvariant="script">S</mi></msubsup><mo>=</mo><mi>Q</mi><msubsup><mi>I</mi><mrow><mi>t</mi><mo>,</mo><msup><mrow><mi mathvariant="script">T</mi></mrow><mo>′</mo></msup></mrow><mi mathvariant="script">S</mi></msubsup></mrow></math>

Graph

. Overall, we obtain:

Graph

Graph

Proof of Theorem 5

(i) We show the existence of nondegenerate Karush–Kuhn–Tucker points of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="script">S</mi></math>

Graph

in a neighborhood of <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo stretchy="false">)</mo></mrow></math>

Graph

<bold> Step 1. </bold> First, we show that for all

Graph

it holds

Graph

. Assume contrarily that

Graph

for some

Graph

. We then have due to T-stationarity, cf. (13):

Graph

Moreover, we have in view of Lemma 1c) an index

Graph

Thus it holds, cf. (13),

Graph

. Due to the assumption on c, we have

Graph

, a contradiction. Hence, we may write:

Graph

Due to NDT6 and NDT3, we may split the other index sets as

Graph

<bold> Step 2. </bold> We consider the auxiliary system of equations

Graph

given by (16)–(22), which mimics stationarity and feasibility. For stationarity we use: 16

Graph

• where

Graph

For feasibility we use: 17

Graph

• 18

Graph

• 19

Graph

• 20

Graph

• 21

Graph

• 22

Graph

In view of feasibility and T-stationarity of

Graph

• for

Graph

, the vector

Graph

solves (16)–(22).

<bold> Step 3. </bold> We consider the Jacobian matrix

Graph

• , where

Graph

the columns of B are given by the vectors:

Graph

and D consists of

Graph

vanishing rows. The remaining rows of D are given by the vectors:

Graph

Additionally we have

Graph

• at

Graph

. Hence, we can apply Theorem 2.3.2 from [[13]], which says that

Graph

is nonsingular if and only if

Graph

• for all

Graph

, the orthogonal complement of the subspace spanned by the columns of B. In view of

Graph

, we check:

Graph

Hence, by means of the implicit function theorem we obtain for any <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>t</mi><mo>></mo><mn>0</mn></mrow></math>

Graph

sufficiently small a solution <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close=")" open="("><mi>t</mi><mo>,</mo><msup><mi>x</mi><mi>t</mi></msup><mo>,</mo><msup><mi>y</mi><mi>t</mi></msup><mo>,</mo><msup><mi>λ</mi><mi>t</mi></msup><mo>,</mo><msup><mi>μ</mi><mi>t</mi></msup><mo>,</mo><msup><mi>σ</mi><mi>t</mi></msup><mo>,</mo><msup><mi>ϱ</mi><mi>t</mi></msup></mfenced></math>

Graph

of the system of equations (16)–(22).

<bold> Step 4. </bold> By choosing t even smaller, if necessary, we can ensure due to continuity reasons as well as NDT2, NDT3, and NDT6 that the following holds:

Graph

, <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><mi>q</mi><mo>∈</mo><mi>Q</mi><mo stretchy="true">\</mo></mrow><msub><mi>Q</mi><mn>0</mn></msub><mfenced close=")" open="("><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mfenced></mrow></math>

Graph

and <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mi>μ</mi><mrow><mn>1</mn><mo>,</mo><mi>q</mi></mrow><mi>t</mi></msubsup><mo>></mo><mn>0</mn></mrow></math>

Graph

, <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>q</mi><mo>∈</mo><msub><mi>Q</mi><mn>0</mn></msub><mfenced close=")" open="("><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mfenced></mrow></math>

Graph

Graph

, <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>i</mi><mo>∈</mo><msub><mi>a</mi><mn>01</mn></msub><mfenced close=")" open="("><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mfenced></mrow></math>

Graph

Graph

, <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mspace width="0.333333em" /><mtext>sgn</mtext><mspace width="0.333333em" /><mfenced close=")" open="("><msubsup><mi>x</mi><mi>i</mi><mi>t</mi></msubsup></mfenced><mo>=</mo><mspace width="0.333333em" /><mtext>sgn</mtext><mspace width="0.333333em" /><mfenced close=")" open="("><msub><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mi>i</mi></msub></mfenced></mrow></math>

Graph

, <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>i</mi><mo>∈</mo><msub><mi>a</mi><mn>10</mn></msub><mfenced close=")" open="("><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mfenced></mrow></math>

Graph

Graph

, <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mi>ϱ</mi><mrow><mn>2</mn><mo>,</mo><mi>i</mi></mrow><mi>t</mi></msubsup><mo><</mo><mn>0</mn></mrow></math>

Graph

Graph

, <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>i</mi><mo>∈</mo><mfenced close="}" open="{"><mn>1</mn><mo>,</mo><mo>...</mo><mo>,</mo><mi>n</mi></mfenced></mrow></math>

Graph

and <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mi>y</mi><mi>i</mi><mi>t</mi></msubsup><mo><</mo><mn>1</mn><mo>+</mo><mi>ε</mi></mrow></math>

Graph

, <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>i</mi><mo>∈</mo><mfenced close="}" open="{"><mn>1</mn><mo>,</mo><mo>...</mo><mo>,</mo><mi>n</mi></mfenced><mrow><mo stretchy="true">\</mo><mi mathvariant="script">E</mi></mrow><mfenced close=")" open="("><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mfenced></mrow></math>

Graph

From here it is straightforward to see that <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close=")" open="("><msup><mi>x</mi><mi>t</mi></msup><mo>,</mo><msup><mi>y</mi><mi>t</mi></msup></mfenced></math>

Graph

is feasible for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="script">S</mi></math>

Graph

and we have:

Graph

, <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="script">E</mi><mfenced close=")" open="("><msup><mi>y</mi><mi>t</mi></msup></mfenced><mo>=</mo><mi mathvariant="script">E</mi><mfenced close=")" open="("><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mfenced></mrow></math>

Graph

, <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="script">N</mi><mfenced close=")" open="("><msup><mi>y</mi><mi>t</mi></msup></mfenced><mo>=</mo><msubsup><mi>a</mi><mn>10</mn><mo>></mo></msubsup><mfenced close=")" open="("><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mfenced></mrow></math>

Graph

, <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="script">O</mi><mfenced close=")" open="("><msup><mi>x</mi><mi>t</mi></msup><mo>,</mo><msup><mi>y</mi><mi>t</mi></msup></mfenced><mo>=</mo><mi mathvariant="normal">∅</mi></mrow></math>

Graph

Graph

Graph

Thus, it holds:

Graph

We rename the multipliers as <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtable><mtr><mtd columnalign="right"><mrow><msubsup><mi>η</mi><mi>i</mi><mrow><mo>≥</mo><mo>,</mo><mi>t</mi></mrow></msubsup><mo>=</mo></mrow></mtd><mtd columnalign="left"><mrow><mrow /><mfenced open="{"><mrow><mtable><mtr><mtd columnalign="left"><mstyle displaystyle="true" scriptlevel="0"><mrow><msubsup><mi>σ</mi><mrow><mn>1</mn><mo>,</mo><mi>i</mi></mrow><mi>t</mi></msubsup><mfrac><mn>1</mn><msub><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mi>i</mi></msub></mfrac><mo>,</mo></mrow></mstyle></mtd><mtd columnalign="left"><mrow><mrow /><mspace width="0.333333em" /><mtext>for</mtext><mspace width="0.333333em" /><mi>i</mi><mo>∈</mo><msup><mrow><mi mathvariant="script">H</mi></mrow><mo>≥</mo></msup><mfenced close=")" open="("><msup><mi>x</mi><mi>t</mi></msup><mo>,</mo><msup><mi>y</mi><mi>t</mi></msup></mfenced><mo>∩</mo><msub><mi>a</mi><mn>01</mn></msub><mfenced close=")" open="("><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mfenced><mo>,</mo></mrow></mtd></mtr><mtr><mtd columnalign="left"><mstyle displaystyle="true" scriptlevel="0"><mrow><mrow /><msubsup><mi>σ</mi><mrow><mn>2</mn><mo>,</mo><mi>i</mi></mrow><mi>t</mi></msubsup><mfrac><mn>1</mn><msub><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mi>i</mi></msub></mfrac><mo>,</mo></mrow></mstyle></mtd><mtd columnalign="left"><mrow><mrow /><mspace width="0.333333em" /><mtext>for</mtext><mspace width="0.333333em" /><mi>i</mi><mo>∈</mo><msup><mrow><mi mathvariant="script">H</mi></mrow><mo>≥</mo></msup><mfenced close=")" open="("><msup><mi>x</mi><mi>t</mi></msup><mo>,</mo><msup><mi>y</mi><mi>t</mi></msup></mfenced><mo>∩</mo><msub><mi>a</mi><mn>10</mn></msub><mfenced close=")" open="("><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mfenced><mo>,</mo></mrow></mtd></mtr><mtr><mtd columnalign="left"><mstyle displaystyle="true" scriptlevel="0"><mrow><mrow /><mfrac><msqrt><mrow><mo>-</mo><msubsup><mi>ϱ</mi><mrow><mn>1</mn><mo>,</mo><mi>i</mi></mrow><mi>t</mi></msubsup><msubsup><mi>ϱ</mi><mrow><mn>2</mn><mo>,</mo><mi>i</mi></mrow><mi>t</mi></msubsup></mrow></msqrt><msqrt><mi>t</mi></msqrt></mfrac><mo>,</mo></mrow></mstyle></mtd><mtd columnalign="left"><mrow><mrow /><mspace width="0.333333em" /><mtext>for</mtext><mspace width="0.333333em" /><mi>i</mi><mo>∈</mo><msup><mrow><mi mathvariant="script">H</mi></mrow><mo>≥</mo></msup><mfenced close=")" open="("><msup><mi>x</mi><mi>t</mi></msup><mo>,</mo><msup><mi>y</mi><mi>t</mi></msup></mfenced><mo>∩</mo><msub><mi>a</mi><mn>00</mn></msub><mfenced close=")" open="("><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mfenced><mo>,</mo></mrow></mtd></mtr><mtr><mtd columnalign="left"><mrow><mrow /><mn>0</mn><mo>,</mo></mrow></mtd><mtd columnalign="left"><mrow><mrow /><mspace width="0.333333em" /><mtext>else,</mtext><mspace width="0.333333em" /></mrow></mtd></mtr></mtable></mrow></mfenced></mrow></mtd></mtr><mtr><mtd columnalign="right"><mrow><mrow /><msubsup><mi>η</mi><mi>i</mi><mrow><mo>≤</mo><mo>,</mo><mi>t</mi></mrow></msubsup><mo>=</mo></mrow></mtd><mtd columnalign="left"><mrow><mrow /><mfenced open="{"><mrow><mtable><mtr><mtd columnalign="left"><mstyle displaystyle="true" scriptlevel="0"><mrow><mo>-</mo><msubsup><mi>σ</mi><mrow><mn>1</mn><mo>,</mo><mi>i</mi></mrow><mi>t</mi></msubsup><mfrac><mn>1</mn><msub><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mi>i</mi></msub></mfrac><mo>,</mo></mrow></mstyle></mtd><mtd columnalign="left"><mrow><mrow /><mspace width="0.333333em" /><mtext>for</mtext><mspace width="0.333333em" /><mi>i</mi><mo>∈</mo><msup><mrow><mi mathvariant="script">H</mi></mrow><mo>≤</mo></msup><mfenced close=")" open="("><msup><mi>x</mi><mi>t</mi></msup><mo>,</mo><msup><mi>y</mi><mi>t</mi></msup></mfenced><mo>∩</mo><msub><mi>a</mi><mn>01</mn></msub><mfenced close=")" open="("><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mfenced><mo>,</mo></mrow></mtd></mtr><mtr><mtd columnalign="left"><mstyle displaystyle="true" scriptlevel="0"><mrow><mrow /><mo>-</mo><msubsup><mi>σ</mi><mrow><mn>2</mn><mo>,</mo><mi>i</mi></mrow><mi>t</mi></msubsup><mfrac><mn>1</mn><msub><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mi>i</mi></msub></mfrac><mo>,</mo></mrow></mstyle></mtd><mtd columnalign="left"><mrow><mrow /><mspace width="0.333333em" /><mtext>for</mtext><mspace width="0.333333em" /><mi>i</mi><mo>∈</mo><msup><mrow><mi mathvariant="script">H</mi></mrow><mo>≤</mo></msup><mfenced close=")" open="("><msup><mi>x</mi><mi>t</mi></msup><mo>,</mo><msup><mi>y</mi><mi>t</mi></msup></mfenced><mo>∩</mo><msub><mi>a</mi><mn>10</mn></msub><mfenced close=")" open="("><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mfenced><mo>,</mo></mrow></mtd></mtr><mtr><mtd columnalign="left"><mstyle displaystyle="true" scriptlevel="0"><mrow><mrow /><mfrac><msqrt><mrow><msubsup><mi>ϱ</mi><mrow><mn>1</mn><mo>,</mo><mi>i</mi></mrow><mi>t</mi></msubsup><msubsup><mi>ϱ</mi><mrow><mn>2</mn><mo>,</mo><mi>i</mi></mrow><mi>t</mi></msubsup></mrow></msqrt><msqrt><mi>t</mi></msqrt></mfrac><mo>,</mo></mrow></mstyle></mtd><mtd columnalign="left"><mrow><mrow /><mspace width="0.333333em" /><mtext>for</mtext><mspace width="0.333333em" /><mi>i</mi><mo>∈</mo><msup><mrow><mi mathvariant="script">H</mi></mrow><mo>≤</mo></msup><mfenced close=")" open="("><msup><mi>x</mi><mi>t</mi></msup><mo>,</mo><msup><mi>y</mi><mi>t</mi></msup></mfenced><mo>∩</mo><msub><mi>a</mi><mn>00</mn></msub><mfenced close=")" open="("><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mfenced><mo>,</mo></mrow></mtd></mtr><mtr><mtd columnalign="left"><mrow><mrow /><mn>0</mn><mo>,</mo></mrow></mtd><mtd columnalign="left"><mrow><mrow /><mspace width="0.333333em" /><mtext>else,</mtext><mspace width="0.333333em" /></mrow></mtd></mtr></mtable></mrow></mfenced></mrow></mtd></mtr><mtr><mtd columnalign="right"><mrow><mrow /><msubsup><mi>ν</mi><mi>i</mi><mi>t</mi></msubsup><mo>=</mo></mrow></mtd><mtd columnalign="left"><mrow><mrow /><mfenced open="{"><mrow><mtable><mtr><mtd columnalign="left"><mstyle displaystyle="true" scriptlevel="0"><mrow><msubsup><mi>σ</mi><mrow><mn>2</mn><mo>,</mo><mi>i</mi></mrow><mi>t</mi></msubsup><mo>,</mo></mrow></mstyle></mtd><mtd columnalign="left"><mrow><mrow /><mspace width="0.333333em" /><mtext>for</mtext><mspace width="0.333333em" /><mi>i</mi><mo>∈</mo><mi mathvariant="script">N</mi><mfenced close=")" open="("><msup><mi>y</mi><mi>t</mi></msup></mfenced><mo>∩</mo><msub><mi>a</mi><mn>10</mn></msub><mfenced close=")" open="("><mover accent="true"><mrow><mi>x</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo>,</mo><mover accent="true"><mrow><mi>y</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mfenced><mo>,</mo></mrow></mtd></mtr><mtr><mtd columnalign="left"><mrow><mrow /><mn>0</mn><mo>,</mo></mrow></mtd><mtd columnalign="left"><mrow><mrow /><mspace width="0.333333em" /><mtext>else.</mtext><mspace width="0.333333em" /></mrow></mtd></mtr></mtable></mrow></mfenced></mrow></mtd></mtr><mtr><mtd columnalign="right"><mrow /></mtd></mtr></mtable></mrow></math> </ephtml>

Graph

Hence, <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close=")" open="("><msup><mi>x</mi><mi>t</mi></msup><mo>,</mo><msup><mi>y</mi><mi>t</mi></msup></mfenced></math>

Graph

fulfills (9). Also, it is straightforward to check that (10) and (11) are fulfilled. Thus, <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close=")" open="("><msup><mi>x</mi><mi>t</mi></msup><mo>,</mo><msup><mi>y</mi><mi>t</mi></msup></mfenced></math>

Graph

is a Karush–Kuhn–Tucker point of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="script">S</mi></math>

Graph

<bold> Step 5. </bold> Moreover, ND1 is satisfied as well in view of Theorem 2. Similar to (10) and (11), ND2 holds at

Graph

. It remains to show ND3, i.e.

Graph

• where

Graph

form a basis of

Graph

, cf. Step 1a from the proof of Theorem 3. Note, that by construction

Graph

is constant for t sufficiently small. Thus, we refer to it as

Graph

. Next, we construct for

Graph

sufficiently small such a basis as follows. First, we choose eigenvectors

Graph

• of

Graph

forming a basis of

Graph

, cf. Step 1b from the proof of Theorem 4. With similar arguments as in Step 2b of the proof of Theorem 4 and by using the implicit function theorem, we find

Graph

• ,

Graph

, still linearly independent. The remaining

Graph

vectors are chosen as follows. Namely, for

Graph

we consider

Graph

As in Step 4 of the proof of Theorem 4, we can find

Graph

. Especially,

Graph

. We note that

Graph

. Using this and the implicit function theorem, we find for t sufficiently small a vector

Graph

. It is then straightforward to check, that

Graph

• and

Graph

• ,

Graph

, indeed form a basis of

Graph

. We continue by considering the following limits with respect to the subsets of

Graph

for any sequence of vectors

Graph

from the constructed base, cf. the definition of

Graph

-multipliers:

Graph

If for some

Graph

• or

Graph

it holds

Graph

, we observe:

Graph

Finally, we calculate as in (15):

Graph

Altogether, we obtain for any basis vector

Graph

• of

Graph

• : 23

Graph

• where

Graph

is a nonzero eigenvalue of

Graph

due to the choice of

Graph

and NDT4. Let us now focus on the basis vectors

Graph

• ,

Graph

. We then have: 24

Graph

This is due to the following reasoning. First,

Graph

is bounded due to the construction of

Graph

, and, moreover,

Graph

. We conclude that ND3 is fulfilled. Additionally, the T-index of

Graph

is equal to the sum of its quadratic index and its biactive index, i.e.

Graph

. In view of (23) and (24), the quadratic index

Graph

• of

Graph

is then exactly m for t sufficiently small. (ii) Next, we elaborate on the uniqueness of Karush–Kuhn–Tucker points

Graph

constructed above.

<bold> Step 6. </bold> As preliminary considerations we show Steps 6.1

Graph

• 6.3.

<bold> Step 6.1. </bold> For all sufficiently small t we show

Graph

First we note, that due to Remark 1 we have

Graph

. Suppose

Graph

. Due to (13), we have

Graph

. Assume

Graph

• . Since

Graph

is a Karush–Kuhn–Tucker point (12),

Graph

. Taking the limit yields a contradiction. Thus,

Graph

. The reverse inclusion follows from Lemma 3d). For the other assertion, we just recall

Graph

, cf. Step 1.

<bold> Step 6.2. </bold> For all sufficiently small t we show

Graph

Lemma 3c) provides

Graph

. Suppose

Graph

. Then, the

Graph

-th row of (9) provides 25

Graph

However, we also have due to T-stationarity of

Graph

that the

Graph

-th row of (5) reads as 26

Graph

The application of Lemma 3a) yields

Graph

for t sufficiently small. Remark 1a) states

Graph

• ,

Graph

. By taking

Graph

in (25) and comparing to (26), we obtain

Graph

In view of NDT3, we know that

Graph

. Thus, we conclude, due to

Graph

• , that

Graph

and, thus, by definition

Graph

. Analogously, we have

Graph

• for all

Graph

. Both assertions then follow.

<bold> Step 6.3. </bold> For all sufficiently small t we show

Graph

• Suppose

Graph

• . The

Graph

-th row of (9) provides again 27

Graph

• The

Graph

-th row of (5) provides 28

Graph

As above, we apply Lemma 3a) and Remark 1a). By taking

Graph

in (27) and comparing to (28), we obtain

Graph

We conclude due to

Graph

and NDT6, i.e.

Graph

, that it holds

Graph

and, thus, by definition

Graph

. Analogously, we have

Graph

• for all

Graph

. We conclude the proof of Step 6.2 by noting that

Graph

. Assume instead there exists

Graph

, then we derive as before

Graph

, a contradiction to NDT6.

<bold> Step 7. </bold> We recall that any Karush–Kuhn–Tucker point

Graph

• of

Graph

with multipliers

Graph

has to fulfill (9)–(11). We define the multipliers

Graph

Let us consider the point

Graph

. In view of Lemma 3 and Lemma 2, we obtain for t sufficiently small

Graph

Due to Step 6, an immediate calculation shows then that

Graph

fulfills equations (16)–(22) for t sufficiently small. However, the inverse function theorem was used in Step 3 to find the solution of this system of equations in the neighborhood of

Graph

. Hence,

Graph

must be unique.

Graph

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References

1 Scholtes S. Convergence properties of a regularization scheme for mathematical programs with complementarity constraints. SIAM J. Optim. 2001; 11: 918-936. 1855214. 10.1137/S1052623499361233 2 Izmailov AF, Solodov MV. Mathematical programs with vanishing constraints: Optimality conditions, sensitivity, and a relaxation method. J. Optim. Theory Appl. 2009; 142: 501-532. 2535113. 10.1007/s10957-009-9517-4 3 Kanzow C, Mehlitz P, Steck D. Relaxation schemes for mathematical programmes with switching constraints. Optim. Methods Softw. 2021; 36: 1223-1258. 4432938. 10.1080/10556788.2019.1663425 4 Lämmel, S, Shikhman, V: Optimality conditions for mathematical programs with orthogonality type constraints. Set-Valued Variat. Anal. 31(9) (2023) 5 Burdakov OP, Kanzow C, Schwartz A. Mathematical programs with cardinality constraints: reformulation by complementarity-type conditions and a regularization method. SIAM J. Optim. 2016; 26: 397-425. 3456393. 10.1137/140978077 6 Bucher M, Schwartz A. Second-order optimality conditions and improved convergence results for regularization methods for cardinality-constrained optimization problems. J. Optim. Theory Appl. 2018; 178: 383-410. 3825630. 10.1007/s10957-018-1320-7 7 Branda M, Bucher M, Červinka M, Schwartz A. Convergence of a Scholtes-type regularization method for cardinality-constrained optimization problems with an application in sparse robust portfolio optimization. Comput. Optim. Appl. 2018; 70: 503-530. 3795255. 10.1007/s10589-018-9985-2 8 Jongen HT, Jonker P, Twilt F. Nonlinear Optimization in Finite Dimensions. 2000: Dordrecht; Kluwer Academic Publishers 9 Lämmel S, Shikhman V. Global aspects of the continuous reformulation for cardinality-constrained optimization problems. Optimization. 2023. 10.1080/02331934.2023.2249014Online first Lämmel S, Shikhman V. Cardinality-constrained optimization problems in general position and beyond. Pure Appl. Funct. Anal. 2023; 8; 4: 1107-1133. 4657026 Červinka M, Kanzow C, Schwartz A. Constraint qualifications and optimality conditions for optimization problems with cardinality constraints. Math. Program. 2016; 160: 353-377. 3555392. 10.1007/s10107-016-0986-6 Günzel H. The structured jet transversality theorem. Optimization. 2008; 57: 159-164. 2384972. 10.1080/02331930701779039 Jongen HT, Meer K, Triesch E. Optimization Theory. 2004: Dordrecht; Kluwer Academic Publishers

By Sebastian Lämmel and Vladimir Shikhman Reported by Author; Author

Result: Extended convergence analysis of the Scholtes-type regularization for cardinality-constrained optimization problems

Further Information

Extended convergence analysis of the Scholtes-type regularization for cardinality-constrained optimization problems

Introduction

Preliminaries

Definition 1

Definition 2

Definition 3

Definition 4

Definition 5

Definition 6

Definition 7

Definition 8

Lemma 1

Theorem 1

Scholtes-type regularization

Definition 9

Theorem 2

Proof

Definition 10

Definition 11

Definition 12

Lemma 2

Proof

Theorem 3

Remark 1

Example 1

Lemma 3

Proof

Theorem 4

Example 2

Remark 2

Lemma 4

Proof

Corollary 1

Proof

Remark 3

Theorem 5

Corollary 2

Proof

Definition 13

Example 3

Conclusions

Acknowledgements

Funding

Declarations

Conflict of interest

Appendix

Proof of Theorem 4

Proof of Theorem 5

Publisher's Note

References

Links

Additional functions