A New Primal-Dual Algorithm for Minimizing the Sum of Three Functions with a Linear Operator

In this paper, we propose a new primal-dual algorithm for minimizing , where f, g, and h are proper lower semi-continuous convex functions, f is differentiable with a Lipschitz continuous gradient, and is a bounded linear operator. The proposed algorithm has some famous primal-dual algorithms for minimizing the sum of two functions as special cases. E.g., it reduces to the Chambolle-Pock algorithm when and the proximal alternating predictor-corrector when . For the general convex case, we prove the convergence of this new algorithm in terms of the distance to a fixed point by showing that the iteration is a nonexpansive operator. In addition, we prove the O(1 / k) ergodic convergence rate in the primal-dual gap. With additional assumptions, we derive the linear convergence rate in terms of the distance to the fixed point. Comparing to other primal-dual algorithms for solving the same problem, this algorithm extends the range of acceptable parameters to ensure its convergence and has a smaller per-iteration cost. The numerical experiments show the efficiency of this algorithm.