Porosity 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 a lower semicontinuous function bounded from below. This paper is concerned with the perturbed optimization problem inf z∈Z {J (z) + x − z }, denoted by (x, J )-inf for x ∈ X. In the case when X is compactly fully 2-convex, it is proved in the present paper 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 = inf z∈Z {J (z) + x − z } is a σ -porous set in X. Furthermore, if X is assumed additionally to be compactly locally uniformly convex, we verify that the set of all points x ∈ X \ Z 0 such that the problem (x, J )-inf fails to be approximately compact, is a σ -porous set in X \ Z 0 , where Z 0 denotes the set of all z ∈ Z such that z ∈ P Z (z). Moreover, a counterexample to which some results of Ni [R.X. Ni, Generic solutions for some perturbed optimization problem in nonreflexive Banach space, J. Math. Anal. Appl. 302 (2005) 417-424] fail is provided.

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

“…These results are in the spirit of the idea due to Blasi, Myjak and Papini in [7]. Extensions to convex sets and generalized approximations of this idea of Blasi, Myjak and Papini can be found in [18][19][20][21][22].…”

“…Cobzas studied in [10] the existence problem in an arbitrary Banach space and proved that if Z is a weakly compact subset of X and J is an upper semicontinuous real-valued functional bounded from above, 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, which extends Lau's result in [23]. For other results on perturbed optimization problems of this kind, one can see, for example, [6,9,10,20].…”

“…The existence results have been applied to optimal control problems governed by partial differential equations, see, for example, [3,4,6,[8][9][10]14,24]. Here we are especially interested in the study of the problem (x, J )-sup; while, for the problem (x, J )-inf, the readers are referred to [6,10,11,20,26]. In [5], it was proved that if X is a reflexive and locally uniformly convex Banach space 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.…”

Categorical properties of solvability for one class of minimization problems1

“…is typical for a closed nonempty subset X of a uniformly convex Banach space E and a bounded below lower semicontinuous functional f X : ® R. The recent results related to the categorical properties of solvability of problem (2) are reviewed in [9].…”

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].…”