Relaxed μ-quasimonotone variational inequalities in Hadamard manifolds

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.