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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Bibliographic Details
Published in:Journal of mathematical analysis and applications Vol. 346; no. 2; pp. 384 - 394
Main Authors: Peng, Li-Hui, Li, Chong, Yao, Jen-Chih
Format: Journal Article
Language:English
Published: San Diego, CA Elsevier Inc 15.10.2008
Elsevier
Subjects:
[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
Online Access:Get full text
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Summary: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