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Statistical Performance of Subgradient Step-Size Update Rules in Lagrangian Relaxations of Chance-Constrained Optimization Models

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Optimization and Applications (OPTIMA 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14395))

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Abstract

Lagrangian relaxation schemes, coupled with a subgradient procedure, are frequently employed to solve chance-constrained optimization models. Subgradient procedures typically rely on step-size update rules. Although there is extensive research on the properties of these step-size update rules, there is little consensus on which rules are most suitable practically; especially, when the underlying model is a computationally challenging instance of a chance-constrained program. To close this gap, we seek to determine whether a single step-size rule can be statistically guaranteed to perform better than others. We couple the Lagrangian procedure with three strategies to identify lower bounds for two-stage chance-constrained programs. We consider two instances of such models that differ in the presence of binary variables in the second-stage. With a series of computational experiments, we demonstrate—in marked contrast to existing theoretical results—that no significant statistical differences in terms of optimality gaps is detected between six well-known step-size update rules. Despite this, our results demonstrate that a Lagrangian procedure provides computational benefit over a naive solution method—regardless of the underlying step-size update rule.

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Acknowledgments

We gratefully acknowledge the compute resources and support provided by the Erlangen Regional Computing Center (RRZE). The authors acknowledge the financial support by the Federal Ministry for Economic Affairs and Energy of Germany in the project METIS (project number 03ET4064).

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Authors and Affiliations

  1. Department of Mathematics, Friedrich-Alexander-Universität Erlangen-Nürnberg, Erlangen, 91058, Germany

    Charlotte Ritter

  2. School of Mathematical Sciences, University of Southampton, Southampton, SO17 1BJ, UK

    Bismark Singh

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  1. Charlotte Ritter
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  2. Bismark Singh
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Correspondence to Bismark Singh .

Editor information

Editors and Affiliations

  1. FRC CSC RAS, Moscow, Russia

    Nicholas Olenev

  2. FRC CSC RAS, Moscow, Russia

    Yuri Evtushenko

  3. University of Montenegro, Podgorica, Montenegro

    Milojica Jaćimović

  4. Krasovsky Institute of Mathematics and Mechanics, Ekaterinburg, Russia

    Michael Khachay

  5. FRC CSC RAS, Moscow, Russia

    Vlasta Malkova

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Data Availability

All our codes and data, as well as Appendix A and B containing our algorithms and computational results, are publicly available at:  https://github.com/charlotteritter/ArticleSubgradient.

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Ritter, C., Singh, B. (2023). Statistical Performance of Subgradient Step-Size Update Rules in Lagrangian Relaxations of Chance-Constrained Optimization Models. In: Olenev, N., Evtushenko, Y., Jaćimović, M., Khachay, M., Malkova, V. (eds) Optimization and Applications. OPTIMA 2023. Lecture Notes in Computer Science, vol 14395. Springer, Cham. https://doi.org/10.1007/978-3-031-47859-8_26

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  • DOI: https://doi.org/10.1007/978-3-031-47859-8_26

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-47858-1

  • Online ISBN: 978-3-031-47859-8

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