Variable Metric Composite Proximal Alternating Linearized Minimization for Nonconvex Nonsmooth Optimization

TL;DR

This paper introduces a novel variable metric proximal alternating linearized minimization algorithm for complex nonconvex nonsmooth optimization problems involving two block variables, with proven convergence and demonstrated effectiveness in MRI reconstruction.

Contribution

The paper proposes the CPALM algorithm for a broad class of nonconvex nonsmooth problems, extending existing methods with convergence guarantees based on the Kurdyka- property.

Findings

Convergence of CPALM to critical points is established.

Numerical results confirm the method's effectiveness in MRI reconstruction.

The algorithm handles complex nonsmooth, nonconvex functions efficiently.

Abstract

In this paper we propose a proximal algorithm for minimizing an objective function of two block variables consisting of three terms: 1) a smooth function, 2) a nonsmooth function which is a composition between a strictly increasing, concave, differentiable function and a convex nonsmooth function, and 3) a smooth function which couples the two block variables. We propose a variable metric composite proximal alternating linearized minimization (CPALM) to solve this class of problems. Building on the powerful Kurdyka-\L ojasiewicz property, we derive the convergence analysis and establish that each bounded sequence generated by CPALM globally converges to a critical point. We demonstrate the CPALM method on parallel magnetic resonance image reconstruction problems. The obtained numerical results shows the viability and effectiveness of the proposed method.