Convergence Analysis of A Proximal Linearized ADMM Algorithm for Nonconvex Nonsmooth Optimization

TL;DR

This paper analyzes the convergence of a generalized proximal linearized ADMM algorithm for nonconvex, nonsmooth optimization problems, establishing boundedness, convergence to critical points, and convergence rates under Kurdyka-Łojasiewicz properties.

Contribution

It introduces a generalized PL-ADMM with variable metric and over-relaxation, providing convergence analysis and rates for nonconvex nonsmooth problems.

Findings

Sequence generated is bounded and converges to critical points.

Under Kurdyka-Łojasiewicz properties, the sequence has finite length and converges.

Convergence rates are established for the proposed algorithm.

Abstract

In this paper, we consider a proximal linearized alternating direction method of multipliers (PL-ADMM) for solving linearly constrained nonconvex and possibly nonsmooth optimization problems. The algorithm is generalized by using variable metric proximal terms in the primal updates and an over-relaxation stepsize in the multiplier update. We prove that the sequence generated by this method is bounded and its limit points are critical points. Under the powerful Kurdyka-{\L ojasiewicz} properties we prove that the sequence has a finite length thus converges, and we drive its convergence rates.