Adaptive Augmented Lagrangian Methods: Algorithms and Practical Numerical Experience

TL;DR

This paper introduces adaptive augmented Lagrangian algorithms with line search and trust region variants, demonstrating their efficiency and reliability in large-scale nonlinear optimization through theoretical guarantees and extensive numerical testing.

Contribution

It presents a novel line search variant of adaptive AL methods with convergence proofs and compares its performance with trust region approaches in practical large-scale problems.

Findings

Adaptive algorithms outperform traditional AL methods in efficiency.

Line search variant achieves global convergence similar to trust region methods.

Numerical tests show robustness on large-scale optimization problems.

Abstract

In this paper, we consider augmented Lagrangian (AL) algorithms for solving large-scale nonlinear optimization problems that execute adaptive strategies for updating the penalty parameter. Our work is motivated by the recently proposed adaptive AL trust region method by Curtis, Jiang, and Robinson [Math. Prog., DOI: 10.1007/s10107-014-0784-y, 2013]. The first focal point of this paper is a new variant of the approach that employs a line search rather than a trust region strategy, where a critical algorithmic feature for the line search strategy is the use of convexified piecewise quadratic models of the AL function for computing the search directions. We prove global convergence guarantees for our line search algorithm that are on par with those for the previously proposed trust region method. A second focal point of this paper is the practical performance of the line search and trust region algorithm variants in Matlab software, as well as that of an adaptive penalty parameter updating strategy incorporated into the Lancelot software. We test these methods on problems from the CUTEst and COPS collections, as well as on challenging test problems related to optimal power flow. Our numerical experience suggests that the adaptive algorithms outperform traditional AL methods in terms of efficiency and reliability. As with traditional AL algorithms, the adaptive methods are matrix-free and thus represent a viable option for solving extreme-scale problems.