Linearized Block-Wise Alternating Direction Method Of Multipliers With Relaxing Proximal Step Size

Convex optimization problems are applied in various fields,such as traffic management,statistical analysis,network economy,etc.Therefore,Efficient algorithm design has always been a hot topic for researchers in the optimization field.Alternating Direction Method of Multipliers(ADMM)is one of the algorithms for solving linear constrained convex optimization problems.It can ensure convergence under the condition of two variables.In order to deal with multi-block problems,multi-block variables can be divided into two groups.Gauss-Seidel format is adopted between groups(using new information in time)and Jacobi format(using old information)is adopted in groups.The subproblems of the algorithm are more troublesome to solve.The predecessors linearized the objective function or objective function and quadratic term of the subproblems and added the proximal point term to simplify the calculation.However,the selection of the proximal factor of the algorithm is limited by the maximum eigenvalue of the constraint matrix of each group of variables,resulting in slow convergence speed.A linearized block-wise alternating direction method of multipliers under the condition of new parameters is proposed to improve the proximal factor in the algorithm,and the convergence rate of the algorithm is greatly accelerated on the premise of keeping the calculation of each step unchanged.The details are organized as follows:In chapter 1,we introduce the research background and its significance of this paper,a brief introduction of Alternating direction method of multipliers and the main research content of this paper.In chapter 2,it introduces the related knowledge and convergence analysis of block-wise alternating direction method in detail.In chapter 3,aiming at the multi-block linear constrained optimization problem with separable objective function,a new algorithm is proposed by improving the blockwise alternating direction multiplier method,and its related properties,predictive correction format and convergence are analyzed.In chapter 4,on the basis of block-wise alternating direction method of multipliers,the objective function and quadratic term of the research problem are linearized simultaneously,the proximity factor is improved,another new algorithm is proposed,and its convergence is analyzed.In chapter 5,it summarizes and prospects the research in this thesis.