Approximate Convexity of Set-Valued Mappings and Variational Inequalities

In this article, we introduce the notion of approximate convexity for set-valued mappings, specifically in the forms of approximate pseudoconvexity and approximate quasiconvexity. These generalizations are motivated by the need to handle optimization problems involving multi-valued operators and vector-valued objective functions, where classical convexity assumptions are too restrictive. We demonstrate that the proposed framework preserves the essential structural features of convex analysis while broadening its applicability. After that, we investigate the introduced definitions through illustrative examples. Furthermore, we consider a set-valued optimization problem and rigorously investigate the relationships among its efficient solutions and the solutions of generalized Minty and Stampacchia variational inequality problems. The results provide a coherent theoretical bridge between optimality conditions and variational inequality formulations for set-valued mappings.