Characterizations of generalized Levitin–Polyak well-posed set optimization problems

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.