Modified proximal technique to decompose variational inequality problems

Abstract

Our study was concerned by a development of an algorithm that decomposes the original variational inequality problem into a series of low-dimensional sub-problems with nonlinear constraints. Also, it is worth to mention that the difficulty here is how can we deal with the nonlinearity of the constraints in our variational inequality model and in the same time how can we obtain a decomposition scheme where the convergence is guaranteed under mild conditions. Our work has reached the following points: 1) Presenting the Variational Inequality Problems (VIP) to the reader with some theoretical results on existence, uniqueness of solutions and convergence of some classical approaches. Also, in chapter 2, a nonexhaustive summary about the link of .VIP's and other mathematical topics was given 2) Presenting a recent algorithm to solve a large scale VIP developed and studied by Yang and his associates (1999-2001). 3) Developing a new algorithm based on auxiliary principle mixed with quadratic and non-quadratic proximal point techniques to solve a VIP with nonlinear constraints.