Research On Generalized Alternating Direction Method Of Multipliers Based On Proximal Term

Alternating direction method of multipliers(ADMM)is an effective tool for solving separable convex optimization problems,and has been widely used in many fields.Compared with the original ADMM,the generalized alternating direction multiplier method has obvious advantages in iteration speed and numerical effects.In recent years,many scholars have continuously deepened and improved their research on ADMM.This paper mainly studies the generalized Peacemen-Rachford(PR)splitting method with indefinite proximal term and the inertial proximal generalized alternating direction method of multipliers for solving two separable convex optimization problems with linear equality constraints.First part,the combination coefficient in the y-subproblem in the generalized PR splitting method is changed,the indefinite proximal term is introduced,and a generalized PR splitting method with indefinite proximal term is proposed.Under some weak conditions,the global convergence of the iterative sequence generated by the proposed algorithm is proved,and the worst-case(46)(1 t)convergence rate in the ergodic sense is established,where t denotes the iteration counter,and a good lower bound of the proximal parameter is obtained,solving the problem of difficult value selection in practical applications.What's more,the effectiveness of the lower bound and the proposed algorithm are further illustrated in numerical experiments.Second part,an inertial proximal PR splitting method recently proposed is an improved method of semi-proximal PR splitting method.Motivated by the idea,based on the original proximal generalized alternating direction method of multipliers(SGADMM),inertial technology is introduced and an inertial proximal generalized alternating direction method of multipliers is proposed.This method employs the current iteration information and the previous iteration information and generate new iteration points,accelerating the convergence of SGADMM.The proposed method is more general,and inertial proximal ADMM,original ADMM and SGADMM can be considered as their special cases.This method also has a large range of relaxation parameters,and it is more conducive to the selection of parameter values in the practical problems.For any inertial sequence,the convergence of the proposed algorithm cannot be guaranteed.Under the simple assumption of the inertial sequence,the global convergence of the method is proved.Furthermore,numerical experiments demonstrate that the algorithm is an effective method.