LEVITIN-POLYAK WELL-POSEDNESS OF MIXED VARIATIONAL INEQUALITIES INVOLVING A BIFUNCTION

Abstract

In this paper, we investigate the existence and Levitin–Polyak (LP) well-posedness of a mixed variational inequality problem involving a bifunction. Sufficient conditions for the existence of solutions are established. We explore the connection between saddle points of the associated Lagrangian and solutions to both the original variational inequality and its Minty counterpart. The study presents several key results on LP well-posedness and generalized LP well-posedness, formulated in terms of the behaviour of the approximate solution sets. A notable aspect of this work is the well-posedness analysis based on the gap function approach. More specifically, sufficient conditions for LP well-posedness of mixed variational inequality problem are provided in terms of the level boundedness of its gap function. Furthermore, an equivalence between the LP well-posedness of the mixed variational inequality problem and that of a related optimization problem is also established.