A proximal parallel splitting method for minimizing sum of convex functions with linear constraints

In this paper, we propose a proximal parallel decomposition algorithm for solving the optimization problems where the objective function is the sum of mm separable functions (i.e., they have no crossed variables), and the constraint set is the intersection of Cartesian products of some simple sets and a linear manifold. The mm subproblems are solved simultaneously per iterations, which are sum of the decomposed subproblems of the augmented Lagrange function and a quadratic term. Hence our algorithm is named as the ‘proximal parallel splitting method’. We prove the global convergence of the proposed algorithm under some mild conditions that the underlying functions are convex and the solution set is nonempty. To make the subproblems easier, some linearized versions of the proposed algorithm are also presented, together with their global convergence analysis. Finally, some preliminary numerical results are reported to support the efficiency of the new algorithms.