Relaxed Projection And Contraction Methods And Selective Projection Methods

This thesis studies two kinds of new methods under the framework of Hilbert space.For the Lipschitz continuous variational inequalities defined on a single set of convex function level,a relaxed projection and contraction method is proposed in this paper.This algorithm maintains the difference between the parameters of the projection step and the correction step in the original projection and contraction method.Further,the projection iterative step replace the projection of the domain with the relaxed projection on the single half-space containing the domain,and the half-space relaxed projection or no projection is used in the correction iteration step,so that the algorithm is more easily implemented,and improved the projection and contraction method proposed by He Bingsheng.For the convex feasibility problems,the multiple-sets split feasibility problems and the strong monotone variational inequality problems defined in the intersection of multiple convex function level sets,the usual algorithms often use projection operators for each level set in each iteration.The algorithms are complex and computationally intensive.When the structure of these level sets is complex,the algorithm is more difficult to implement.For this reason,the paper proposes a selective projection methods.In each iteration one of the levels is selected according to a certain rule.The iteration format only involves the projection of the level set or the half-space relaxation projection of the level set.The main advantage of the new algorithm is that the algorithm is simple in format,easier to implement,and less computationally burdensome.The details of this article are as follows:Firstly,in order to increase the feasibility of the original projection and contraction method,a relaxed projection and contraction method is proposed.The weak convergence theorem of the new algorithm is proved and the convergence rate of the algorithm is estimated.Secondly,we propose selective projection methods for solving the convex feasibility problems,the multiple-sets split feasibility problems and the strong monotone variational inequality problems,and the convergence of the algorithm is proved.In addition,Take the problem of the convex feasibility problems and strong monotone variational inequality asexamples,the results of numerical simulation experiments show the effectiveness of the selective projection methods.Thirdly,based on the idea of selective projection methods,it is generalized to solve the common fixed point problem of finite nonexpansive mappings family,the more general multiple-sets split feasibility problems and the variational inequality defined on common fixed point sets problems.