An Accelerated Inexact Dampened Augmented Lagrangian Method for Linearly-Constrained Nonconvex Composite Optimization Problems

TL;DR

This paper introduces an accelerated inexact dampened augmented Lagrangian method for efficiently solving linearly-constrained nonconvex composite optimization problems, with theoretical convergence guarantees and numerical validation.

Contribution

It develops a novel AIDAL algorithm combining inexact subproblem solving, dampening, and adaptive penalty updates, advancing optimization techniques for nonconvex constrained problems.

Findings

Converges to an approximate stationary point in ${ m O}(\varepsilon^{-5/2}\lograc{1}{\varepsilon})$ iterations.

Demonstrates computational efficiency through numerical experiments.

Provides theoretical analysis under mild assumptions.

Abstract

This paper proposes and analyzes an accelerated inexact dampened augmented Lagrangian (AIDAL) method for solving linearly-constrained nonconvex composite optimization problems. Each iteration of the AIDAL method consists of: (i) inexactly solving a dampened proximal augmented Lagrangian (AL) subproblem by calling an accelerated composite gradient (ACG) subroutine; (ii) applying a dampened and under-relaxed Lagrange multiplier update; and (iii) using a novel test to check whether the penalty parameter of the AL function should be increased. Under several mild assumptions involving the dampening factor and the under-relaxation constant, it is shown that the AIDAL method generates an approximate stationary point of the constrained problem in ${\cal O}(\varepsilon^{-5/2}\log\varepsilon^{-1})$ iterations of the ACG subroutine, for a given tolerance $\varepsilon>0$. Numerical experiments are also given to show the computational efficiency of the proposed method.