A new primal-dual method for minimizing the sum of three functions with a linear operator

In this paper, we propose a new primal-dual algorithm for minimizing $f(\vx)+g(\vx)+h(\vA\vx)$, where $f$, $g$, and $h$ are convex functions, $f$ is differentiable with a Lipschitz continuous gradient, and $\vA$ is a bounded linear operator. It has some famous primal-dual algorithms for minimizing the sum of two functions as special cases. For example, it reduces to the Chambolle-Pock algorithm when $f=0$ and a primal-dual fixed-point algorithm in [P. Chen, J. Huang, and X. Zhang, A primal-dual fixed-point algorithm for convex separable minimization with applications to image restoration, Inverse Problems, 29 (2013), p.025011] when $g=0$. In the general convex case, we prove the $o(1/k)$ convergence rate 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 ergodic and non-ergodic convergence rates in terms of a 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 new algorithm by comparing with other primal-dual algorithms.