Necessary conditions for upper semicontinuity in parametric semi-infinite programming (Q762068)

We treat semi-infinite optimization problems: Minimize p(x) subject to \(x\in {\mathbb{R}}^ m\), and a(t,x)\(\leq b(t)\) for all \(t\in T\), where T is a \(\sigma\)-compact topological space, and p, a, b are suitable (- \(\infty,\infty)\)-valued functions on \({\mathbb{R}}^ m\), respectively. Linear, convex, and quasi-convex semi-infinite programming are included in this concept. The main results of this paper are on the necessity of the compactness of the set of feasible points for (a,b), and the set of \(\phi\)-optimal solutions for (p,a,b) for the (Hausdorff) upper semicontinuity of the feasible set-mapping in (a,b), and the \(\phi\)- optimal solution-mapping in (p,a,b), respectively (where the parameter sets are provided with a suitable topology). Some more special results complete the paper.