Parallel Alternating Direction Primal-Dual (PADPD) Algorithm for Multi-Block Centralized Optimization

In this article, a centralized two-block separable convex optimization with equality constraint and its extension to multi-block optimization are considered. The first fully parallel primal-dual discrete-time algorithm called Parallel Alternating Direction Primal-Dual (PADPD) is proposed. In the algorithm, the primal variables are updated in an alternating fashion like Alternating Direction Method of Multipliers (ADMM). The algorithm can handle non-smoothness of objective functions with strong convergence. Unlike existing discrete-time algorithms such as Method of Multipliers (MM), ADMM, Parallel ADMM, Bi-Alternating Direction Method of Multipliers (Bi-ADMM), and Primal-Dual Fixed Point (PDFP) algorithms, all primal and dual variables in the proposed algorithm are updated independently. Therefore, the time complexity of the algorithm can be significantly reduced. It is shown that the rate of convergence of the algorithm for quadratic or linear cost functions is exponential or linear under suitable assumptions. The algorithm can be directly extended to any finite multi-block optimization without further assumptions while preserving its convergence. PADPD algorithm not only can compute more iterations (since it is fully parallel) for the same time-step but it is also possible that PADPD algorithm can have a faster convergence rate than that of ADMM. Finally, two numerical examples are provided in order to show advantage of PADPD algorithm.