Two Algorithms For Solving Separable Convex Optimization Problems

Optimization problem is an important tool in study of mathematics,engineering science and management science,and has been widely used in economics,mathematical programming,network traffic optimization,image processing.Consequently,designing effective algorithm for solving these problem is a hot research topic.This dissertation mainly studied the convex optimization problem with three separable operator,and based on the variational inequality theory,projection algorithm theory and the theory of alternating direction propose some improved algorithms,The specific content of this dissertation is as follows:1.In chapter 1,we firstly introduce the general variational inequalities and some basic properties of the variational inequality,and some projection mapping basic defi-nition.Then,we introduce the research status and application of alternating direction multiplier method and parallel splitting algorithm for solving separable convex opti-mization problems.We also give the motivation and the main research work of this thesis.2.In Chapter 2,we study the application of alternating direction multiplier method in solving convex optimization problems with linear constraints and objective function is the sum of three independent function without coupling variables.Since two functions are not easy to optimize,we introduce projection gradient alternating direction method and prove its global convergence,convergence rate in an ergodic sense and nonergence sense.3.In Chapter 3,combining with the parallel splitting algorithm and adjacent point algorithm,we introduce a new adjacent parallel splitting algorithm and prove its the convergence.for the numerical analysis,we give a practical example to show the effective and superiority of this new algorithm.