Iteratively Linearized Reweighted Alternating Direction Method of Multipliers for a Class of Nonconvex Problems

TL;DR

This paper introduces a reweighted ADMM algorithm for nonconvex, nonsmooth problems in signal processing and machine learning, ensuring convergence and computational efficiency.

Contribution

The paper proposes a novel reweighted ADMM that guarantees convergence for a class of challenging nonconvex, nonsmooth problems, with all subproblems being convex and solvable.

Findings

Algorithm converges globally to a critical point.

Numerical experiments demonstrate high efficiency.

All subproblems are convex and easy to solve.

Abstract

In this paper, we consider solving a class of nonconvex and nonsmooth problems frequently appearing in signal processing and machine learning research. The traditional alternating direction method of multipliers encounters troubles in both mathematics and computations in solving the nonconvex and nonsmooth subproblem. In view of this, we propose a reweighted alternating direction method of multipliers. In this algorithm, all subproblems are convex and easy to solve. We also provide several guarantees for the convergence and prove that the algorithm globally converges to a critical point of an auxiliary function with the help of the Kurdyka-{\L}ojasiewicz property. Several numerical results are presented to demonstrate the efficiency of the proposed algorithm.