Relaxed μ -quasimonotone variational inequalities in Hadamard manifolds

Abstract

In this article, we introduce relaxed μ-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 μ-quasiconvexity and show that the Upper Dini and Clarke–Rockafellar subdifferentials of a relaxed μ-quasiconvex function is relaxed μ-quasimonotone. Under relaxed μ-quasiconvexity in the nondifferentiable sense, we establish the connection between the Stampacchia variational inequality problem and a nonsmooth constrained optimization problem. © 2019, Springer Nature Switzerland AG.