Convergence Analysis Of Two Types Of ADMM With Positive-indefinite Proximal Regularization

In recent years,the alternating direction method of multipliers(ADMM)has been wellrecognized as a versatile approach for multi-block linearly constrained convex minimization models with separable structures.Especially,the proximal types of ADMM which involves the proximal regularization of ADMM's subproblems has been well-studied and widely used.However,the existing results on the convergence of the proximal types of ADMM inevitably require the positive definiteness of the related proximal matrix,which usually will small step sizes and the overall convergence rate of this type method as reported in the literature.In this paper,we propose two types of modified positive-indefinite proximal linearized ADMM with a larger step size for updating the dual variable for solving two-block linearly constrained separable convex programing.We investigate the internal relationships between the step size coefficient and the penalty coefficient to identify the convergence of the modified ADMM.Then the convergence of the modified ADMM and their convergence rate measured by the iteration complexity are established in ergodic sense.Finally,we demonstrate the computational performance of the modified ADMM by numerical experiments.