Adaptive augmented Lagrangian methods: algorithms and practical numerical experience

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 et al. [An adaptive augmented Lagrangian method for large-scale constrained optimization, Math. Program. 152 (2015), pp. 201–245.]. 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 large-scale problems.

Keywords:

• nonlinear optimization
• non-convex optimization
• large-scale optimization
• augmented Lagrangians
• matrix-free methods
• steering methods

AMS Subject Classifications:

• 49M05
• 49M15
• 49M29
• 49M37
• 65K05
• 65K10
• 90C06
• 90C30
• 93B40

Acknowledgements

We would like to thank Sven Leyffer and Victor Zavala from Argonne National Laboratory for providing us with the AMPL [1,21] files required to test the optimal power flow problems described in Section 3.1.4.

Disclosure

No potential conflict of interest was reported by the authors.

Notes

1. We convert all general inequality constraints to equality constraints by using slack variables. Other approaches, for example, [3–5], use an AL function defined for the inequality constraints instead of introducing additional slack variables.

Additional information

Funding

Frank E. Curtis was supported by Department of Energy grant [DE–SC0010615], Nicholas I. M. Gould was supported by Engineering and Physical Sciences Research Council grant [EP/I013067/1], Hao Jiang and Daniel P. Robinson were supported by National Science Foundation grant [DMS-1217153].