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

Abstract

In this article, we analyze Levitin–Polyak (LP) well-posedness of a mixed variational inequality problem involving a bifunction. Sufficient criteria are derived that assert the solution existence. 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 establishes key results on LP well-posedness and generalized LP well-posedness, characterizing them through the behavior of approximate solution sets. A notable aspect of this work is the well-posedness analysis based on the gap function approach. In particular, we establish suitable criteria for the LP well-posedness of the mixed variational inequality problem by examining the level boundedness of its associated gap function. Furthermore, the LP well-posedness of the mixed variational inequality problem is reduced to verifying the well-posedness of a related optimization problem.