Papers › A new primal-dual algorithm for minimizing the sum of three functions with a linear operator
A new primal-dual algorithm for minimizing the sum of three functions with a linear operator
Ming Yan
In this paper, we propose a new primal-dual algorithm for minimizing f(x) + g(x) + h(Ax), where f, g, and h are proper lower semi-continuous convex functions, f is differentiable with a Lipschitz continuous gradient, and A 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 f = 0 and the proximal alternating predictor-corrector when g = 0. 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.
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