Perturbed optimization in Banach spaces III: Semi-infinite optimization

This paper is devoted to the study of perturbed semi-infinite optimization problems, i.e., minimization over ]R 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 (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() of continuous real-valued functions. Key words, sensitivity analysis, marginal function, approximate solutions, directional constraint qualification, semi-infinite programming, epilimits AMS subject classifications. 46N10, 47H19, 49K27, 49K40, 58C15, 90C31