An ADMM-like Algorithm For Solving Extended Separable Convex Optimization Problems With Linear Constraints

The alternating direction method of multipliers(ADMM)is widely used in var-ious fields,it is efficient for two-block convex optimization with linear equality constraints.The structured extensive separable convex optimization with linear constraints is also considered in various practical applications.After introducing a slack variable,it can be translated to a three-block problem with equality con-straints.However,the direct extension of ADMM with three separable operators is not necessarily convergent.For tackling the lack of convergence of the direct exten-sion of ADMM,some researchers have proposed two main strategies.The first is improving the condition of the model,the second is to change the direct extension method.In this paper,we propose a novel ADMM-like method for solving the three-block problem which translated from the original two-block condition with inequality constraints.We just do some slight changes on the updating of the slack variable on the basic of the direct extension of ADMM.Specifically,these changes include adding a proximal term to z-subproblem and correcting the slack variable in a simple form.For the convergence analysis,this method can be interpreted as a prediction correction algorithm.Finally,the global convergence of this new method can be easily obtained within the analytic framework of contraction-type methods.We further estimate the worst-case convergence rate in an ergodic sense.Some numerical results show the efficiency of the proposed method.