Scalarization and Well-Posedness for Set Optimization Problems Involving General Set Less Relations

The aim of this paper is to investigate the scalarization and well-posedness for set optimization problems where the solution concept is given by the lower set less relation induced by general sets. First, we propose three kinds of well-posedness for set optimization problems via general sets and study their relationships. Moreover, some sufficient conditions for these kinds of well-posedness for set optimization problems via free-disposal set are obtained under the Hausdorff P-continuity of the objective mapping of the set optimization problem, rather than the continuity in the sense of Berge. By employing a nonlinear scalarization technique by means of the oriented distance function, corresponding scalar optimization problems are established, and the relationship between solutions of the set optimization problem where the solution concept is given by the lower set less relation induced by co-radiant sets and solutions of the corresponding scalarized problem is also discussed. Furthermore, we also give some characterization of well-posedness for set optimization problem through two scalar optimization problems. Some examples are given to illustrate the main results.