Two-step Fixed-point proximity algorithms for multi-block separable convex problems

TL;DR

This paper introduces a novel two-step fixed-point proximity algorithm for multi-block separable convex problems, ensuring convergence and efficiency, and demonstrates its effectiveness through numerical experiments on sparse MRI reconstruction.

Contribution

The paper develops a new convergent two-step fixed-point proximity algorithm specifically designed for multi-block separable convex problems, with proven convergence and practical efficiency.

Findings

The proposed 2SFPPA algorithm converges globally.

The algorithm achieves an O(1/k) convergence rate.

Numerical experiments show high efficiency in sparse MRI problems.

Abstract

Multi-block separable convex problems recently received considerable attention. This class of optimization problems minimizes a separable convex objective function with linear constraints. The algorithmic challenges come from the fact that the classic alternating direction method of multipliers (ADMM) for the problem is not necessarily convergent. However, it is observed that ADMM outperforms numerically many of its variants with guaranteed theoretical convergence. The goal of this paper is to develop convergent and computationally efficient algorithms for solving multi-block separable convex problems. We first characterize the solutions of the optimization problems by proximity operators of the convex functions involved in their objective function. We then design a two-step fixed-point iterative scheme for solving these problems based on the characterization. We further prove convergence of the iterative scheme and show that it has O(1/k) convergence rate in the ergodic sense and the sense of the partial primal-dual gap, where k denotes the iteration number. Moreover, we derive specific two-step fixed-point proximity algorithms (2SFPPA) from the proposed iterative scheme and establish their global convergence. Numerical experiments for solving the sparse MRI problem demonstrate the numerical efficiency of the proposed 2SFPPA.