A Predictor-Corrector Algorithm For Solving Separable Convex Optimization Problems

In recent years,the separable convex optimization problems with linear constraints has been widely used in many fields,such as engineering,management and so on.How to solve this kind of separable convex programming problem with linear constraints has attracted the attention of many scholars.For this kind of problem,when the number of variables is more than two,the direct application of the alternating direction method can not guarantee its convergence in theory.Therefore,the researchers proposed some other methods based on the alternating direction method,such as predictor-corrector algorithm.In this paper,we study the solution of linearly constrained separable convex optimization problems with three separable variables.In the framework of the predictor-corrector algorithm,three new algorithms are presented to solve the problem.Their convergence of the three algorithms are proved,and numerical examples are given to illustrate the effectiveness of the three algorithms.The contents of this paper are as follows:In Chapter 1,we introduce the research background and the present situation of the problem of linear constrained separable convex optimization problems,and the main structure of the paper is given.In Chapter 2,we introduce some related knowledge,including some notations,defi-nitions and theorems.In Chapter 3,we propose a new partial parallel predictor-corrector algorithm based on the existing prediction-correction algorithm.The convergence of the algorithm is proved.Finally,numerical examples are given to show the effectiveness of the proposed algorithm.In Chapter 4,based on the idea of partial parallel predictor-corrector algorithm pro-posed in the third chapter,we propose a completely parallel predictor-corrector algorithm,and prove the convergence of the algorithm.Finally,the effectiveness of the proposed al-gorithm is demonstrated by numerical examples.In Chapter 5,The existing algorithm with projection correction step is improved,and a new partial parallel predictor-corrector algorithm with projection correction is presented.Then the convergence of the algorithm is proved.Finally,a numerical example is given to illustrate the effectiveness of the algorithm.In Chapter 6,we summarize the research work of this paper and make a prospect for the future research work.