Convergence Analysis of a Relative-type Inexact Preconditioned Proximal ALM for Convex Nonlinear Programming

TL;DR

This paper rigorously analyzes the convergence properties of a relative-type inexact preconditioned proximal augmented Lagrangian method for convex nonlinear programming, establishing global and asymptotic convergence rates.

Contribution

It provides the first rigorous proof of global convergence and superlinear rates for rip$^2$ALM, enhancing theoretical understanding of proximal augmented Lagrangian methods.

Findings

Proves global convergence of rip$^2$ALM.

Establishes asymptotic superlinear convergence rate.

Derives global ergodic convergence rates for feasibility and objective residuals.

Abstract

This article investigates the convergence properties of a relative-type inexact preconditioned proximal augmented Lagrangian method (rip$^2$ALM) for convex nonlinear programming, a fundamental class of optimization problems with broad applications in science and engineering. Inexact proximal augmented Lagrangian methods have proven to be highly effective for solving such problems, owing to their attractive theoretical properties and strong practical performance. However, the convergence behavior of the relative-type inexact preconditioned variant remains insufficiently understood. This work aims to reduce this gap by rigorously establishing the global convergence of the sequence generated by rip$^2$ALM and proving its asymptotic (super)linear convergence rate under standard assumptions. In addition, we derive the global ergodic convergence rate with respect to both the primal feasibility violation and the primal objective residual, thereby offering a more comprehensive understanding of the overall performance of rip$^2$ALM. These results deepen our theoretical understanding of the family of proximal augmented Lagrangian methods and motivate their development for practical, large-scale structured application problems.