Scalarization, convergence and well-posedness in set optimization

In this paper, we introduce the notions of weak $ m $-minimal approximate solutions and $ m $-minimal approximate solutions for constrained set optimization problems, based on a novel set order relation that involves the Minkowski difference. We derive scalarization results for the sets of weak $ m $-minimal approximate solutions and $ m $-minimal approximate solutions in the context of set optimization. Utilizing these scalarizations, we analyze the Painlevé-Kuratowski convergence properties of both classes of approximate solutions. Furthermore, new notions of well-posedness for set optimization problems are proposed, and relationships between these notions are rigorously established. Finally, we establish equivalences between the well-posedness of set optimization problems and their scalar counterparts through carefully constructed optimization frameworks.