Well-posedness of a class of perturbed optimization problems in Banach spaces

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 i...

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Vydáno v:	Journal of mathematical analysis and applications Ročník 346; číslo 2; s. 384 - 394
Hlavní autoři:	Peng, Li-Hui, Li, Chong, Yao, Jen-Chih
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	San Diego, CA Elsevier Inc 15.10.2008 Elsevier
Témata:	[formula omitted]-subset Calculus of variations and optimal control Exact sciences and technology Functional analysis Kadec space Mathematical analysis Mathematics Numerical analysis Numerical analysis. Scientific computation Numerical methods in mathematical programming, optimization and calculus of variations Perturbed optimization problem Sciences and techniques of general use Strictly convex space Well-posedness Kadec space Strictly convex space G δ -subset Perturbed optimization problem Well-posedness Lower bound Perturbed optimization problem;Strictly convex space;Kadec space;Well-posedness;Gδ-subset Semicontinuous function Mathematical analysis Optimization method Banach space Application Optimization
ISSN:	0022-247X, 1096-0813
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Popis
Shrnutí:	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 .
ISSN:	0022-247X 1096-0813
DOI:	10.1016/j.jmaa.2008.05.069