Stability for a Class of Nonlinear Optimization Problems and Applications

Abstract

The aim of this chapter is to give a unified approach to some problems in nonlinear optimization using asymptotic cones, recession functions and asymptotically compact sets. Thus we establish a stability result for a class of nonconvex programming problems which turns out to be equivalent to Dedieu’s criterion for the closedeness of the image of a closed set by a multifunction. Also we obtain a formula for the recession function of the marginal function for the first time. This formula seems to be important and new also in the finite dimensional case. The convex version of the stability result is used to reobtain formulae for the conjugates, ϵ-subdifferentials and recession functions of some convex functions, results which are comparable with those of McLinden. It is also shown that in some cases one can perturbe the objective function of a family of convex problems such that the resulting problems have optimal solutions; the behaviour of the values of these perturbed problems and their solutions is also investigated. Another result establishes the relationship between conically compact sets introduced by Isac and Thera and asymptotic cones.