Improved proximal ADMM with partially parallel splitting for multi-block separable convex programming

For a type of multi-block separable convex programming raised in machine learning and statistical inference, we propose a proximal alternating direction method of multiplier with partially parallel splitting, which has the following nice properties: (1) to alleviate the weight of the proximal terms, the restrictions imposed on the proximal parameters are relaxed substantively; (2) to maintain the inherent structure of the primal variables x(i) (i = 1, 2, . . . , m), the relaxation parameter gamma is only attached to the update formula of the dual variable lambda. For the resulted method, we establish its global convergence and worst-case O(1/t) convergence rate in an ergodic sense, where t is the iteration counter. Finally, three numerical examples are given to illustrate the theoretical results obtained.