Generic Well-Posedness of Optimization Problems and the Banach-Mazur Game

Let X be a completely regular topological space. Denote, as usual, by C(X) the family of all bounded continuous real-valued functions in X. The space C(X) equipped with the sup-norm ||f||∞ = sup{| f(x)|: x ∈ X}, f ∈ C(X), becomes a Banach space. Each f ∈ C(X) determines a minimization problem: find x0 ∈ X with f(x 0) = inf {f(x) : x ∈ X} =: inf (X, f). We designate this problem by (X, f). Among the different properties of the minimization problem (X, f) the following ones are of special interest in the theory of optimization: (a) (X, f) has a solution (existence of the solution); (b) the solution set for (X, f) is a singleton (uniqueness of the solution); (c) if f(x*) is close to inf (X, f), then x* is a good approximation of the solution of (X, f) (stability of the solution—see bellow the precise definition).

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