Convergence analysis of a relaxed inertial alternating minimization algorithm with applications

TL;DR

This paper introduces a new inertial three-block alternating minimization algorithm for convex problems with linear constraints, proving its convergence and demonstrating its efficiency through numerical experiments.

Contribution

It develops a novel inertial extension of the three-block AMA with convergence guarantees, improving upon existing methods like ADMM.

Findings

The proposed algorithm converges under mild conditions.

Numerical experiments show the algorithm's efficiency in SPCP problems.

The new method simplifies subproblems compared to three-block ADMM.

Abstract

The alternating direction method of multipliers (ADMM) is a popular method for solving convex separable minimization problems with linear equality constraints. The generalization of the two-block ADMM to the three-block ADMM is not trivial since the three-block ADMM is not convergence in general. Many variants of three-block ADMM have been developed with guarantee convergence. Besides the ADMM, the alternating minimization algorithm (AMA) is also an important algorithm for solving the convex separable minimization problem with linear equality constraints. The AMA is first proposed by Tseng, and it is equivalent to the forward-backward splitting algorithm applied to the corresponding dual problem. In this paper, we design a variant of three-block AMA, which is derived by employing an inertial extension of the three-operator splitting algorithm to the dual problem. Compared with three-block ADMM, the first subproblem of the proposed algorithm only minimizing the Lagrangian function. As a by-product, we obtain a relaxed algorithm of Davis and Yin. Under mild conditions on the parameters, we establish the convergence of the proposed algorithm in infinite-dimensional Hilbert spaces. Finally, we conduct numerical experiments on the stable principal component pursuit (SPCP) to verify the efficiency and effectiveness of the proposed algorithm.