Some Therotical Research Of Hybrid Proximal Extragradient Method

The monotone inclusion problems is a class of important problems in optimization.The hybrid proximal extra-gradient algorithm is a class of inaccurate proximal point algorithm for solving monotone inclusion problems.In this paper,based on hybrid proximal extragradient method,we proposed some improved and generalized hybrid proximal extra-gradient algorithms.In the first chapter,we summarized latest progress and research status of hybrid proximal extragradient method,and some definitions and theorems are introduced.In the second chapter,we propose a new inertial hybrid inexact proximal point algorithm for solving monotone inclusion problems.Under certain conditions,we establish the weak convergence of the point sequence generated by the algorithm,the O((?))non-asymptotic global convergence rate and the O(1/k)ergodic iterative complexity result of the proposed inertial hybrid inexact proximal point method.As an application,we propose and analyze three special cases:the inertial proximal contraction algorithm for solving monotone inclusions problems with monotone and Lipschitz continuous operators,and the inertial projection-proximal point algorithm for solving generalized variational inequalities with monotone operators,and the inertial inexact scaled partial inverse method for solving primal-dual problem with maximal monotone operators.And we obtain the convergence,non-asymptotic global convergence rate and ergodic iterative complexity of the corresponding algorithm under the farmework of the inertial hybrid inexact proximal point algorithm.The results of this paper generalize and improve the corresponding conclusions in the literature.Last,we apply our new inertial ADMM method to LASSO problem,and some preliminary numerical experiments are reported to illustrate the advantages of the new method.In the third chapter,we propose a new inertial hybrid proxiaml extragradient method for solving monotone inclusion problems.We obtain the weak convergence and the non-asymptotic global convergence rate of the inertial hybrid proxiaml extragradient method.In the framework of inertial hybrid proxiaml extragradient method,we propose and analyze the convergemce and non-asymptotic global convergence rate of an inertial Tseng’s forward-backward method,and an inertial inexact Spingarn’s partial inverse method.