Well-posedness of a class of perturbed optimization problems in Banach spaces - Peng, Li-Hui; Li, Chong; Yao, Jen-Chih (Journal of mathematical analysis and applications, 2008)
Let X be a Banach space and Z a nonempty subset of X. Let J : Z → R be a lower semicontinuous function bounded from below and p ⩾ 1 . This paper is concerned with the perturbed optimization problem of finding z 0 ∈ Z such that ‖ x − z 0 ‖ p + J ( z 0 ) = inf z ∈ Z { ‖ x − z ‖ p + J ( z ) } , which is denoted by min J ( x , Z ) . The notions of the J-strictly convex with respect to Z and of the Kadec with respect to Z are introduced and used in the present paper. It is proved that if X is a Kadec Banach space with respect to Z and Z is a closed relatively boundedly weakly compact subset, then the set of all x ∈ X for which every minimizing sequence of the problem min J ( x , Z ) has a converging subsequence is a dense G δ -subset of X ∖ Z 0 , where Z 0 is the set of all points z ∈ Z such that z is a solution of the problem min J ( z , Z ) . If additionally p > 1 and X is J-strictly convex with respect to Z, then the set of all x ∈ X for which the problem min J ( x , Z ) is well-posed is a dense G δ -subset of X ∖ Z 0 .