Well-posedness of a class of perturbed optimization problems in Banach spaces |
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Authors: | Li-Hui Peng Jen-Chih Yao |
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Institution: | a Department of Mathematics, Zhejiang Gongshang University, Hangzhou 310018, PR China b Department of Mathematics, Zhejiang University, Hangzhou 310027, PR China c Department of Applied Mathematics, National Sun Yat-Sen University, Kaohsiung, Taiwan, ROC |
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Abstract: | 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 z0∈Z such that ‖x−z0p‖+J(z0)=infz∈Z{‖x−zp‖+J(z)}, which is denoted by minJ(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 minJ(x,Z) has a converging subsequence is a dense Gδ-subset of X?Z0, where Z0 is the set of all points z∈Z such that z is a solution of the problem minJ(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 minJ(x,Z) is well-posed is a dense Gδ-subset of X?Z0. |
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Keywords: | Perturbed optimization problem Strictly convex space Kadec space Well-posedness _method=retrieve& _eid=1-s2 0-S0022247X08005659& _mathId=si18 gif& _pii=S0022247X08005659& _issn=0022247X& _acct=C000069490& _version=1& _userid=6211566& md5=1a8850078304a66857591f19bf37c7f7')" style="cursor:pointer Gδ-subset" target="_blank">" alt="Click to view the MathML source" title="Click to view the MathML source">Gδ-subset |
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