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.