Iteration-complexity of an inner accelerated inexact proximal augmented Lagrangian method based on the classical Lagrangian function

TL;DR

This paper analyzes the iteration complexity of an accelerated inexact proximal augmented Lagrangian method for nonconvex constrained optimization, showing it converges efficiently under certain conditions and demonstrating strong practical performance.

Contribution

It establishes the iteration complexity of the IAIPAL method based on classical Lagrangian functions for nonconvex problems, including complexity bounds without initial feasibility assumptions.

Findings

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

Complexity bounds hold without requiring a feasible initial point.

Numerical results confirm the practical efficiency of the proposed method.

Abstract

This paper establishes the iteration-complexity of an inner accelerated inexact proximal augmented Lagrangian (IAIPAL) method for solving linearly-constrained smooth nonconvex composite optimization problems that is based on the classical augmented Lagrangian (AL) function. More specifically, each IAIPAL iteration consists of inexactly solving a proximal AL subproblem by an accelerated composite gradient (ACG) method followed by a classical Lagrange multiplier update. Under the assumption that the domain of the composite function is bounded and the problem has a Slater point, it is shown that IAIPAL generates an approximate stationary solution in ${\cal O}(\varepsilon^{-5/2}\log^2 \varepsilon^{-1})$ ACG iterations where $\varepsilon>0$ is a tolerance for both stationarity and feasibility. Moreover, the above bound is derived without assuming that the initial point is feasible. Finally, numerical results are presented to demonstrate the strong practical performance of IAIPAL.