Porosity of perturbed optimization problems in Banach spaces

Let X be a Banach space and Z a nonempty closed subset of X .L etJ : Z → R be a lower semicontinuous function bounded from below. This paper is concerned with the perturbed optimization problem infz∈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 z0 ∈ Z such that J( z0) +� x − z0 �= infz∈Z{ J( z) +� x − z�} is a σ -porous set in X .F urthermore, ifX 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 ,w hereZ 0 denotes the set of all z ∈ Z such that z ∈ PZ(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.

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