Existence and porosity for a class of perturbed optimization problems in Banach spaces

Abstract

Let X be a Banach space and Z a nonempty closed subset of X. Let J : Z → R be an upper semicontinuous function bounded from above. This paper is concerned with the perturbed optimization problem sup z∈Z {J (z) + x − z }, which is denoted by (x, J )-sup. We shall prove in the present paper that if Z is a closed boundedly relatively weakly compact nonempty subset, then the set of all x ∈ X for which the problem (x, J )-sup has a solution is a dense G δ -subset of X. In the case when X is uniformly convex and J is bounded, we will show that the set of all points x in X for which there does not exist z 0 ∈ Z such that J (z 0 ) + x − z 0 = sup z∈Z {J (z) + x − z } is a σ -porous subset of X and the set of all points x ∈ X \ Z 0 such that there exists a maximizing sequence of the problem (x, J )-sup which has no convergent subsequence is a σ -porous subset of X \ Z 0 , where Z 0 denotes the set of all z ∈ Z such that z is in the solution set of (z, J )-sup.

Nearest and farthest points in spaces of curvature bounded below

“…For more developments and extensions in this direction, the readers are referred to [2,3,7,8,11,[22][23][24][25][26][27][28][29][31][32][33] and the surveys [12,30].…”

Well-posedness of a class of perturbed optimization problems in Banach spaces

“…In the case when p = 1, Baranger in [2] proved that if X is a uniformly convex Banach space then the set of all x ∈ X for which the problem min J (x, Z ) has a solution is a dense G δ -subset of X , which clearly extends Stechkin's results in [30] on the best approximation problem. Since then, this problem has been studied extensively, see for example [6,8,20,28]. In particular, Cobzas extended in [9] Baranger's result to the setting of reflexive Kadec Banach space; while Ni relaxed in [27] the reflexivity assumption made in Cobzas' result. For the general case when p > 1, this kind of perturbed optimization problems is only founded to be studied by Bidaut in [6].…”