Treffer: Radial Epiderivatives and Asymptotic Functions in Nonconvex Vector Optimization: Radial epiderivatives and asymptotic functions in nonconvex vector optimization

The author introduces the notions of lower and upper radial epiderivatives for nonconvex vector-valued functions and points out some important properties especially in connection with the characterization of solutions of nonconvex vector optimization problems. Let \(f: X\to Y\) be a vector-valued function where \(X\) and \(Y\) are real normed spaces and \(Y\) is ordered by a closed convex and pointed cone \(P\subset Y\). For a given point \(x^0\in\text{dom\,}f\) and a direction \(u\in X\), the lower and upper radial epiderivative of \(f\) are defined by \[ \begin{aligned} \underline D^R_e f(x^0; u) &= \liminf_{v\to u}\,\inf_{t> 0}\,{f(x^0+ tv)- f(x^0)\over t},\\ \overline D^R_e f(x^0; u) &= \limsup_{v\to u}\, \sup_{t> 0}\, {f(x^0+ tv)- f(x^0)\over t}.\end{aligned} \] Many geometrical and algebraic properties of these differentiability notions are given. Using the radial cone and the interiorly radial cone to a set \(C\) according to \[ R(C; x^0)= \overline{\bigcup_{t>0} t(C- x^0)}\quad\text{and}\quad R^i(C; x^0)= \text{int}\Biggl(\bigcap_{t> 0} t(C- x^0)\Biggr), \] it is shown that the epigraphs and hypographs of the introduced derivatives are closely connected with the radial cone and interiorly radial cone to the epigraph and hypograph of the function \(f\). The results are used for the construction of optimality conditions for unconstrained nonconvex vector optimization problems.