Second Order Analysis in Semi-Infinite Programming

Abstract

Consider the problem of minimizing the function

$$ f(x) = \mathop{{\max }}\limits_{{y \in Y(x)}} \;{f_{o}}(x,y) $$

((1))

where f _o is a continuous function on X×Y (both X and Y being finite dimensional spaces), Y(x) a compact-valued map from X into Y and either (a) Y does not depend on x and f _o is continuous together with its first and second order derivatives with respect to x, or (b) Y(x) = {y: f _i .(x, y) ≤ 0, i = 1,..., k; f _i (x, y) = 0, i = k + 1,..., m} and all functions f _i , i = 0,....., m are of class C ².