Metrically Levitin–Polyak well-posed parametric set optimization problem

In this article, concepts of metrically pointwise and metrically global Levitin–Polyak (LP) well-posed parametric set optimization problems are introduced. Characterizations of metrically pointwise and metrically global LP well-posed parametric set optimization problems are scrutinized in detail. Connections between metrically pointwise LP well-posed parametric set optimization problems and a approximate solution map are established. Connections between metrically global LP well-posed parametric set optimization problems and the approximate solution map are also established. The role of upper semicontinuity and compactness of the approximate solution map are also analysed for a parametric set optimization problem to be metrically pointwise and global LP well-posed. The role of excess function in the approximate solution map is also scrutinized for a parametric set optimization problem to be metrically pointwise LP well-posed. Relationships between pointwise LP well-posedness and metrically pointwise LP well-posedness are derived. Relationships between global LP well-poseness and metrically global LP well-posedness are also analysed. All the links are properly established to prove the necessary and sufficient conditions.