On scaled stopping criteria for a safeguarded augmented Lagrangian method with theoretical guarantees

This paper discusses the use of a stopping criterion based on the scaling of the Karush–Kuhn–Tucker (KKT) conditions by the norm of the approximate Lagrange multiplier in the ALGENCAN implementation of a safeguarded augmented Lagrangian method. Such stopping criterion is already used in several nonl...

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Bibliographic Details
Published in:Mathematical programming computation 2022-03, Vol.14 (1), p.121-146
Main Authors: Andreani, R., Haeser, G., Schuverdt, M. L., Secchin, L. D., Silva, P. J. S.
Format: Article
Language:English
Subjects:
Algorithms
Convergence
Criteria
Full Length Paper
Kuhn-Tucker method
Lagrange multiplier
Mathematics
Mathematics and Statistics
Mathematics of Computing
Nonlinear programming
Normality
Operations Research/Decision Theory
Optimization
Theory of Computation
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Summary:This paper discusses the use of a stopping criterion based on the scaling of the Karush–Kuhn–Tucker (KKT) conditions by the norm of the approximate Lagrange multiplier in the ALGENCAN implementation of a safeguarded augmented Lagrangian method. Such stopping criterion is already used in several nonlinear programming solvers, but it has not yet been considered in ALGENCAN due to its firm commitment with finding a true KKT point even when the multiplier set is not bounded. In contrast with this view, we present a strong global convergence theory under the quasi-normality constraint qualification, that allows for unbounded multiplier sets, accompanied by an extensive numerical test which shows that the scaled stopping criterion is more efficient in detecting convergence sooner. In particular, by scaling, ALGENCAN is able to recover a solution in some difficult problems where the original implementation fails, while the behavior of the algorithm in the easier instances is maintained. Furthermore, we show that, in some cases, a considerable computational effort is saved, proving the practical usefulness of the proposed strategy.
ISSN:1867-2949
1867-2957
DOI:10.1007/s12532-021-00207-9