Study On The Step Sizes Of The Proximal Admm With Double Multipliers Updatings

The symmetric version of alternating direction method of multipliers(ADMM)is an application of Peaceman-Rachford splitting method.The original symmetric AD-MM is empirical,and its convergence is theoretically not guaranteed.Recently He et al.(2016)take different step sizes to update the multiplier in symmetric ADMM and prove its convergence.Glowinski’s larger step size can be chosen in Lagrange-multiplier-updating steps make the method become more flexible.In this paper,we add positive semidefinite proximal terms to the initial variable updating subproblem-s,and show that we also can choose Glowinski’s larger step size for the one of the Lagrange-multiplier-updating steps at each iteration.We prove the proximal ADM-M is convergent and establish the worst-case O(1/t)convergence rate in the ergodic sense.The numerical experiment on an wide range collection of problems illustrate the advantages of choosing larger step sizes and the effectiveness of the method.