On The Convergence Analysis Of Proximal-based Symmetric Alternating Directions Methods Of Multipliers With Large Step Size

We study the convergence and the numerical performance of proximal-term-based symmetric alternating directions method of multipliers with large step size.The alter-nating direction method of multipliers updates the dual variable A two times at each iteration.So that alternating directions method of multipliers can be seen as special case of symmetric alternating direction method of multipliers.At each iteration,the computation of updating the dual variable is small,while the computation of solving the subproblems of x.y is large.So if we can increase the number of updates to the dual variable to reduce the number of solving the subproblems of x.y,it is possible to improve the efficiency of the method.Because symmetric alternating directions method of multipliers has attractive performance for solving the convex programming problem while theoretically its convergence is not guaranteed.Therefore we consider the convergence analysis of the symmetric alternating direction method of multipliers by introducing the step size of variables.Prior to this,He et al.has proved the convergence of the symmetric alternating directions method of multipliers by employing a suitable step size.Base on this,we ex-pand the step size in symmetric alternating directions method of multipliers.We prove the global convergence of the symmetric alternating direction methods of multipliers under this situation.In addition,in order to make the method more flexible,in this paper,we introduce semi-proximal term for each subproblems in the iteration.More known results on alternating direction method of multipliers literature turn out to be special cases of our discussion.The numerical tests demonstrate the efficiency of the proposed method with large step size.