Scalarization of Levitin–Polyak well-posedness in vector optimization using weak efficiency

In this paper, we study Levitin–Polyak (LP in short) well-posedness of a vector optimization problem (P) in the vectorial sense as well as scalar sense and establish a relation between them. Sufficiency conditions for LP well-posedness for both the notions have been discussed. Further, we use the notion of approximate solutions of scalarized problems associated with problem (P) to establish some other sufficiency criteria for LP well-posedness. This is achieved by establishing sufficiency conditions in terms of closedness and upper semicontinuity of approximate solution map and upper Hausdorff convergence of sequences of approximate solution sets. Finally, we establish the Painlevé–Kuratowski set-convergence of a sequence of optimal solution sets of scalarized problems to the weakly efficient solution set of (P).