Treffer: A Central Cutting Plane Algorithm for Convex Semi-Infinite Programming Problems: A central cutting plane algorithm for convex semi-infinite programming problems

Summary: The central cutting plane algorithm for linear semi-infinite programming (SIP) is extended to nonlinear convex SIP of the form \(\min\{f(x)\mid x\in H, g(x,t)\leq 0\) for all \(t\in S\}\). Under differentiability assumptions that are weaker than those employed in superlinearly convergent algorithms, a linear convergence rate is established that has additional important features. These features are the ability to (i) generate a cut from any violated constraint; (ii) invoke efficient constraint-dropping rules for management of linear programming (LP) subproblem size; (iii) provide an efficient grid management scheme to generate cuts and ultimately to test feasibility to a high degree of accuracy, as well as to provide an automatic grid refinement for use in obtaining admissible starting solutions for the nonlinear system of first-order conditions; and, (iv) provide primal and dual (Lagrangian) SIP feasible solutions in a finite number of iterations. Numerical tests are provided on a collection of problems that have appeared in the literature including some moderately sized problems from complex approximation theory.