Contraction Methods Based On Relaxed PPA

In this paper, we chiefly consider using contraction methods based on relaxed PPA for solving convex optimization problems with linear constraints. And we utilize related knowledge of variational inequalities to prove the convergence of the algorithms which are proposed in this paper. PPA is a class of classic algorithms for solving variational inequality problems and its rate of convergence is generally linear. The whole paper is organized as follows:Chapter 1 introduces variational inequalities, PPA, and relaxed PPA, and states the main research of this paper.Chapter 2 introduces the preliminaries of variational equalities which include pro-jection mapping and its basic properties, monotone operators and convex functions, the equivalent projection equation of variational equalities, and three fundamental inequal-ities.Chapter 3 presents Dual-Primal relaxed PPA based contraction method, including prediction step and correction step in each iteration.Chapter 4 proves the convergence of the proposed methods. First, we prove the generated sequence{uκ} is Fejer monotone. Second, we prove the generated sequence {uκ} converges to a solution of variational inequality. Chapter 5 gives a specific examples of numerical experiments of the proposed algorithms, and compares them with another method. The numerical results show that the proposed algorithms in the paper are effective.The last part concludes all the paper.