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

Let X be a Banach space and Z a nonempty closed subset of X. Let be an upper semicontinuous function bounded from above. This paper is concerned with the perturbed optimization problem supz∈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 Gδ-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 z0∈Z such that J(z0)+‖x−z0‖=supz∈Z{J(z)+‖x−z‖} is a σ-porous subset of X and the set of all points x∈X∖Z0 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∖Z0, where Z0 denotes the set of all z∈Z such that z is in the solution set of (z,J)-sup.