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Polyak Minorant Method for Convex Optimization

Author

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  • Nikhil Devanathan

    (Stanford University)

  • Stephen Boyd

    (Stanford University)

Abstract

In 1963 Boris Polyak suggested a particular step size for gradient descent methods, now known as the Polyak step size, that he later adapted to subgradient methods. The Polyak step size requires knowledge of the optimal value of the minimization problem, which is a strong assumption but one that holds for several important problems. In this paper we extend Polyak’s method to handle constraints and, as a generalization of subgradients, general minorants, which are convex functions that tightly lower bound the objective and constraint functions. We refer to this algorithm as the Polyak Minorant Method (PMM). It is closely related to cutting-plane and bundle methods.

Suggested Citation

  • Nikhil Devanathan & Stephen Boyd, 2024. "Polyak Minorant Method for Convex Optimization," Journal of Optimization Theory and Applications, Springer, vol. 203(3), pages 2263-2282, December.
    • Handle: RePEc:spr:joptap:v:203:y:2024:i:3:d:10.1007_s10957-024-02412-7
    DOI: 10.1007/s10957-024-02412-7
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    References listed on IDEAS

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    1. Wanyou Cheng & Donghui Li, 2012. "An Active Set Modified Polak–Ribiére–Polyak Method for Large-Scale Nonlinear Bound Constrained Optimization," Journal of Optimization Theory and Applications, Springer, vol. 155(3), pages 1084-1094, December.
    2. Damek Davis & Dmitriy Drusvyatskiy & Kellie J. MacPhee & Courtney Paquette, 2018. "Subgradient Methods for Sharp Weakly Convex Functions," Journal of Optimization Theory and Applications, Springer, vol. 179(3), pages 962-982, December.
    3. W. Ackooij & A. Frangioni & W. Oliveira, 2016. "Inexact stabilized Benders’ decomposition approaches with application to chance-constrained problems with finite support," Computational Optimization and Applications, Springer, vol. 65(3), pages 637-669, December.
    4. Yurii Nesterov, 2018. "Lectures on Convex Optimization," Springer Optimization and Its Applications, Springer, edition 2, number 978-3-319-91578-4, December.
    5. R. E. Marsten & W. W. Hogan & J. W. Blankenship, 1975. "The B oxstep Method for Large-Scale Optimization," Operations Research, INFORMS, vol. 23(3), pages 389-405, June.
    6. Lemaréchal, C. & Nemirovskii, A. & Nesterov, Y., 1995. "New variants of bundle methods," LIDAM Reprints CORE 1166, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    7. Guanghui Lan & Zhiqiang Zhou, 2020. "Algorithms for stochastic optimization with function or expectation constraints," Computational Optimization and Applications, Springer, vol. 76(2), pages 461-498, June.
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