An Improvement On Line Search Strategy In The Projection And Contraction Method For Variational Inequality Problems

Variational inequality problem(VIP) is a basic problem in operational research, at the same time, economics, ecology, engineering science and finance and many fields have a wide range of applications. Therefore, since the 1960 s, the variational inequality problem especially its numerical calculation method has received wide attention of many researchers. Such as some known methods including Newton method, alternating directions method, near point method and interior-point method, neural network and projection methods, among them, the projection and contraction method with the characteristics of simple easy to operate, but also got the favour of many scholars. All these researches emerge endlessly,the Projection and Contraction Method put forward by Bingsheng He is one of the category.The characteristic of projection and contraction method is that each iteration computation are generally simple calculation of some functions to the feasible set of projection. And the main idea of this article is to carry on the deep research and attempt to construct a new projection and contraction method to solve the variational inequality problems. Specific work is to improve the existing projection and contraction method. By the exploration on the search direction of Bingsheng He study to consider the selection of step. By the premise of falling in the original direction, to get a efficient step-size. Find a new projection and contraction method to solve the variational inequality problem.The content:The first chapter is mainly the introduction. First of all, carry on the simple introduction of the emergence of the variational inequality problem, development, and other background knowledge. Next presents basic concepts and conclusion of a variational inequality problem and the projection on the convex set. Then, introduces several common methods for solving variational inequality problems. Finally, give the arrangement of the content at the end of this chapter in this paper.The second chapter firstly gives some inequalities about variational inequality problem and the basic idea of the projection and contraction methods proposed by Bingsheng He from the NanJing University. Secondly, combining with the paper of my senior Jinlong Wang, analysis the mechanism of projection and contraction method,propose a new projection and contraction method to improve the line search strategy, prove the global convergence of the new algorithm under the assumption of the original problem has a solution and vector-valued function F monotone. Finally, compared two kinds of methods of reference before and after the improvement through several examples in order to verify the improved new method is more efficient.The third chapter is overall analysis and evaluation of this article, and prospect of the next step research work.