Most of the Optimization Problems have Unique Solution

Abstract

Let X be a compact metric space and C(X) be the space of all continuous real-valued functions in X. Every pair (A, f), where A belongs to the set 2^X of all closed subsets of X and f is from C(X), determines a (constrained) minimization problem: min { f(y): y ∈ A } (find x A at which f attains its minimum over A). Suppose that 2^X is endowed with the Hausdorff metric and C(X) is topologized by the usual uniform convergence norm. We prove that there is a dense G_δ-subset G of 2^X C(X) such that every minimization problem (A, f) from G has unique solution, i.e. the set { x ∈ A: f(x) = min { f(y) : y ∈ A } } consists of only one point for each pair (A, f) outside some first Baire category subset of 2^X×C(X).