An inertial primal‐dual fixed point algorithm for composite optimization problems

Abstract

In this paper, we consider an inertial primal‐dual fixed point algorithm (IPDFP) to compute the minimizations of the sum of a non‐smooth convex function and a finite family of composite non‐smooth convex functions, each one of which is composed of a non‐smooth convex function and a bounded linear operator. This is a full splitting approach, in the sense that non‐smooth functions are processed individually via their proximity operators. The convergence of the IPDFP is obtained by reformulating the problem to the sum of three non‐smooth convex functions. Furthermore, we propose a preconditioning technique for the IPDFP. The key idea of the preconditioning technique is that the constant iterative parameters are updated self‐adaptively in the iteration process. What's more, we also give a simple and easy way to choose the diagonal preconditioners while the convergence of the iterative algorithms is maintained. This work brings together and notably extends several classical splitting schemes, like the primal‐dual method proposed by Chambolle and Pock, and the recent proximity algorithms of Charles et al. designed for the /TV image denoising model. The iterative algorithm is used for solving non‐differentiable convex optimization problems arising in image processing.