Approximate Solutions and Levitin-Polyak Well-Posedness for Set Optimization Using Weak Efficiency

The present study is devoted to define a new notion of approximate weak minimal solution based on a set order relation introduced by Karaman et al. (Positivity 22(3):783-802, 2018) for a constrained set optimization problem. Sufficient conditions have been found for the closedness of minimal solution sets. Using the Painleve-Kuratowski convergence, the stability aspects of the approximate weak minimal solution sets are discussed. Further, a notion of Levitin-Polyak well-posedness for the set optimization problem is introduced. Sufficiency criteria and some characterizations of the above defined well-posedness are established. An alternative approach to obtain robust solutions for uncertain vector optimization problems is discussed as an application.