Perturbed Optimization in Banach Spaces III: Semi-Infinite Optimization

Abstract

This paper is devoted to the study of perturbed semi-infinite optimization problems, i.e., minimization over $\er^n$ with an infinite number of inequality constraints. We obtain the second-order expansion of the optimal value function and the first-order expansion of approximate optimal solutions in two cases: (i) when the number of binding constraints is finite and (ii) when the inequality constraints are parametrized by a real scalar. These results are partly obtained by specializing the sensitivity theory for perturbed optimization developed in part I (cf. [SIAM J. Control Optim., 34 (1996), pp. 1151--1171]) and deriving specific sharp lower estimates for the optimal value function which take into account the curvature of the positive cone in the space $C(\Omega)$ of continuous real-valued functions.