The Convergence Properties of Infeasible Inexact Proximal Alternating Linearized Minimization

TL;DR

This paper analyzes the convergence of an infeasible inexact variant of the PALM algorithm for block-structured optimization, providing theoretical guarantees and demonstrating efficiency through numerical experiments.

Contribution

It introduces a convergence analysis for PALM-I, including a surrogate sequence and inexact criterion, establishing stationarity and convergence rates.

Findings

PALM-I converges to stationary points under certain conditions.

The method shows improved CPU time in quantum physics and frictional contact problems.

Theoretical analysis addresses nonmonotonicity due to infeasibility.

Abstract

The proximal alternating linearized minimization method (PALM) suits well for solving block-structured optimization problems, which are ubiquitous in real applications. In the cases where subproblems do not have closed-form solutions, e.g., due to complex constraints, infeasible subsolvers are indispensable, giving rise to an infeasible inexact PALM (PALM-I). Numerous efforts have been devoted to analyzing feasible PALM, while little attention has been paid to PALM-I. The usage of PALM-I thus lacks theoretical guarantee. The essential difficulty of analyses consists in the objective value nonmonotonicity induced by the infeasibility. We study in the present work the convergence properties of PALM-I. In particular, we construct a surrogate sequence to surmount the nonmonotonicity issue and devise an implementable inexact criterion. Based upon these, we manage to establish the stationarity of any accumulation point and, moreover, show the iterate convergence and the asymptotic convergence rates under the assumption of the Lojasiewicz property. The prominent advantages of PALM-I on CPU time are illustrated via numerical experiments on problems arising from quantum physics and 3D anisotropic frictional contact.