Characterizations of generalized Levitin–Polyak well-posed set optimization problems
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In this paper, we introduce the notion of generalized Levitin–Polyak (in short gLP) well-posedness for set optimization problems. We provide some characterizations of gLP well-posedness in terms of the upper Hausdorff convergence and Painlevé–Kuratowski convergence of a sequence of sets of approximate solutions, and in terms of the upper semicontinuity and closedness of an approximate solution map. We obtain some equivalence relationships between the gLP well-posedness of a set optimization problem and the gLP well-posedness of two corresponding scalar optimization problems. Also, we give some other characterizations of gLP well-posedness by two extended forcing functions and the Kuratowski noncompactness measure of the set of approximate solutions. Finally we show that certain cone-semicontinuous and cone-quasiconvex set optimization problems are gLP well-posed.
KeywordsLevitin–Polyak well-posedness Set optimization Upper semicontinuity Huasdorff convergence Painlevé–Kuratowski convergence Cone-semicontinuity Cone-quasiconvexity
The author is grateful to the anonymous referees for their valuable comments on the first version of the paper.
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