Preconditioned Douglas-Rachford type primal-dual method for solving composite monotone inclusion problems with applications

Abstract

This paper is concerned with the monotone inclusion involving the sum of a finite number of maximally monotone operators and the parallel sum of two maximally monotone operators with bounded linear operators. To solve this monotone inclusion, we first transform it into the formulation of the sum of three maximally monotone operators in a proper product space. Then we derive two efficient iterative algorithms, which combine the partial inverse method with the preconditioned Douglas-Rachford splitting algorithm and the preconditioned proximal point algorithm. Furthermore, we develop an iterative algorithm, which relies on the preconditioned Douglas-Rachford splitting algorithm without using the partial inverse method. We carefully analyze the theoretical convergence of the proposed algorithms. Finally, in order to demonstrate the effectiveness and efficiency of these algorithms, we conduct numerical experiments on a novel image denoising model for salt-andpepper noise removal. Numerical results show the good performance of the proposed algorithms.

Primal-Dual Splitting Algorithms for Solving Structured Monotone Inclusion with Applications

“…The reason is that many convex minimization problems arising in image processing, statistical learning, and economic management can be modelled by such monotone inclusion problems. Based on the perspective of operator splitting algorithms, these primal-dual splitting algorithms can be roughly divided into four categories: (i) Forward-backward splitting type [1][2][3][4]; (ii) Douglas-Rachford splitting type [5][6][7]; (iii) Forward-backward-forward splitting type [8][9][10][11][12]; and (iv) Projective splitting type [13][14][15][16][17]. In 2014, Becker and Combettes [11] first studied the following structured monotone inclusion problem: Problem 1.…”