Relaxed $$\varvec{\mu }$$-quasimonotone variational inequalities in Hadamard manifolds

Abstract

In this article, we introduce relaxed $$ \mu $$ -quasimonotone set-valued vector field on Hadamard manifolds and prove the existence of solutions of the Stampacchia variational inequality for such mappings. We also present the notion of relaxed $$ \mu $$ -quasiconvexity and show that the Upper Dini and Clarke–Rockafellar subdifferentials of a relaxed $$ \mu $$ -quasiconvex function is relaxed $$ \mu $$ -quasimonotone. Under relaxed $$ \mu $$ -quasiconvexity in the nondifferentiable sense, we establish the connection between the Stampacchia variational inequality problem and a nonsmooth constrained optimization problem.