Variational Inequality Systems And Parallel Subgradient Outer Gradient Methods In Hadamard Manifolds

In Hadamard manifold,let the VIP and SVI represent a variational inequality problem and a variational inequality system,respectively,where the SVI consists of two variational inequalities with mutually symmetric structure.Two parallel algorithms are designed to solve the SVI using subgradient extragradient method.Each algorithm consists of two parts with mutually symmetric structure,and some properties of the iterative sequences are proved.In this paper,the results extend and improve some important theorems in recent literatures.The structure of the paper is as follows.The first chapter summarizes the scholars' research on variational inequality and variational inequality system,and puts forth the main research problems of this paper.Chapter two reviews some preliminary concepts,lemmas and propositions in Riemannian geometry.In Chapter 3,we introduce and analyze the first parallel subgradient extragradient rule to solve the variational inequalities system and variational inequality problem,and prove that if the underlying vector field is monotone,the approximate sequences constructed by these algorithms converge to a solution.In Chapter 4,the second parallel subgradient extragradient rule is introduced and analyzed,and the convergence of sequences generated by the algorithm is proved.The fifth chapter is the summary and prospect.