A Linearized Block-by-Block Alternating Direction Multiplier Method With New Parameter Conditions

Alternating Direction Multiplier Method(ADMM)is an effective method forsolving linear constrained optimization problems with two variables,which is widely used in information science,statistics and other fields.However,if this method is directly extended to the case of multiple variables,the algorithm will not converge without proper assumptions.Due to the important application value of this algorithm in related fields,the research on multi-block ADMM algorithm has been increasing in recent years.The block-by-block ADMM algorithm proposed by He Bingsheng and others is a more efficient method.Firstly,the method divides multi-block variables into two groups.When solving sub-problems,Gauss-Seidel format is used between groups(using new information in time),and Jacobi format is used between blocks(using old information).The algorithm has the characteristics of fast convergence of Gauss-Seidel scheme and parallel computation of Jacobi scheme.The one-step computational complexity is low and the global convergence is guaranteed.However,when the original problem has no special structure,it may be difficult to solve the sub-problem of the algorithm.Based on the linearization technique,Henderen et al.proposed a strategy to simplify the calculation of sub-problems by linearizing the sub-problems and increasing the adjacent point terms,which shortened the calculation time.However,the factor selection of the adjacent points is usually limited by the maximum eigenvalue of each group of variable constraint matrices,which makes the convergence speed of the algorithm slow.On the basis of ADMM algorithm,this paper proposes a linearized block-by-block ADMM algorithm with new parameter conditions,which improves the proximity factor of the algorithm proposed by Handren et al.It accelerates the convergence speed of the algorithm while keeping the amount of calculation unchanged at each step,and proves the convergence of the new algorithm.The contents and framework of this paper are as follows:In Chapter 1,mainly introduces the background of the paper and the research process of ADMM algorithm.In Chapter 2,some preliminary knowledge needed for the new algorithm and BADMM algorithm are introduced.In Chapter 3,we introduce a new algorithm,namely the improved linearized block-by-block ADMM algorithm.In Chapter 4,the convergence analysis of the new algorithm is given.In Chapter 5,numerical experiments are carried out to compare the two algorithms and verify their convergence.In Chapter 6,summarizes the research contents of this paper and prospects the research of ADMM algorithm.