Two Kinds Of Separable Lagrangian Function Methods For Minimizing The Sum Of Three Convex Functions

Separable convex optimization problems with linear constraint widely exits in many important fields such as science, engineering, management, etc. In recent years,significant progress has been made in numerical algorithms for solving these problems,many effective methods based on parallel splitting augmented Lagrangian function methods, proximal point algorithms and predictor-corrector alternating direction methods etc. have been developed.In this thesis, we proposed two approximate proximal point algorithms for the linearly constrained convex optimization problem where the objective function is the sum of three convex functions without coupled variables. That is the new predictor-corrector proximal multiplier method and the parallel splitting augmented Lagrangian function method which is based on proximal point. When the alternating direction method is directly extended to solve the problem with the objective function being the sum of three separable convex functions, the global convergence can not be guaranteed. In order to have global convergence property and good numerical performance, we use the ideas of both correction step and proximal point algorithm to construct the algorithms.The thesis is organized as follows.In Chapter 1, Firstly,we give the development of the methods for solving linearly constrained convex optimization problems. Secondly,we have respectively introduced the existing results on separable Lagrangian function methods,which are to solve convex optimization problems with the objective function being the sum of two convexfunctions and the sum of three convex functions as well as with linear constraints.In Chapter 2, we have proposed a new predictor-corrector proximal multiplier method about the linearly constraint convex optimization problem with the objective function is the sum of three separable convex functions. This method is based on the idea of the predictor-corrector proximal multiplier method and proximal point algorithms. Under some assumptions, we have proved global convergence and linear convergence rate of the new predictor-corrector proximal multiplier method.In Chapter 3, on the basis of the ideas about the parallel splitting augmented Lagrangian method and proximal point algorithm, we have proposed a new parallel splitting augmented Lagrangian function method with proximal point algorithm, and its convergence has been proved.In Chapter 4, as for the two new proposed methods, we have made numerical experiment and also have explained the validity of the two methods.Finally, some conclusion are drawn in the last chapter.