In this paper, we study a scalarization function and introduce well-posedness notions for set optimization problems. Some scalarization results are given for the sets of weak $ m $-minimal solutions and $ m $-minimal solutions for set optimization problems. Furthermore, we introduce novel concepts of well-posedness specifically tailored for the set optimization problem. We establish connections among three distinct types of well-posedness and explore the relationships that exist between the well-posedness of set optimization problems and that of scalar optimization problems.

In this paper we investigate the stability of solution sets for set optimization problems via improvement sets. Some sufficient conditions for the upper semicontinuity, lower semicontinuity, and compactness of E-minimal solution mappings are given for parametric set optimization under some suitable conditions. We also give some examples to illustrate our main results.

In this paper, we consider the strictly G-preinvex functions introduced by Antczak (J. Comput. Appl. Math. 217:212-226, 2008). The relationships between semistrictly G-preinvex functions and strictly G-preinvex functions, G-preinvex functions strictly G-preinvex functions are investigated under mild assumptions. Our results improve and extend the existing ones in the literature.

In (J. Inequal. Appl. 2006:9532, 2006), Peng and Zhu discussed interrelations among D-preinvexity, D-semistrict preinvexity, and D-strict preinvexity for vector-valued functions. In this note, we show that the same results or even more general ones can be obtained under weaker assumptions. We also give a new characterization of D-preinvexity and D-semistrict preinvexity under mild conditions.

The current study focuses on exploring the stability of solution sets pertaining to set optimization problems, particularly with regard to the set order relation outlined by Karaman et al. 2018. Sufficient conditions are provided for the lower semicontinuity, upper semicontinuity, and compactness of m-minimal solution mappings in parametric set optimization, where the involved set-valued mapping is Lipschitz continuous.

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.