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Gradient Method with Step Adaptation

Author

Listed:
  • Vladimir Krutikov

    (Institute of Informatics and Telecommunications, Reshetnev Siberian State University of Science and Technology, 31 Krasnoyarskii Rabochii Prospekt, 660037 Krasnoyarsk, Russia
    Department of Applied Mathematics, Kemerovo State University, 6 Krasnaya Street, 650043 Kemerovo, Russia)

  • Elena Tovbis

    (Institute of Informatics and Telecommunications, Reshetnev Siberian State University of Science and Technology, 31 Krasnoyarskii Rabochii Prospekt, 660037 Krasnoyarsk, Russia)

  • Svetlana Gutova

    (Department of Applied Mathematics, Kemerovo State University, 6 Krasnaya Street, 650043 Kemerovo, Russia)

  • Ivan Rozhnov

    (Institute of Informatics and Telecommunications, Reshetnev Siberian State University of Science and Technology, 31 Krasnoyarskii Rabochii Prospekt, 660037 Krasnoyarsk, Russia
    Laboratory “Hybrid Methods of Modeling and Optimization in Complex Systems”, Siberian Federal University, 79 Svobodny Prospekt, 660041 Krasnoyarsk, Russia)

  • Lev Kazakovtsev

    (Institute of Informatics and Telecommunications, Reshetnev Siberian State University of Science and Technology, 31 Krasnoyarskii Rabochii Prospekt, 660037 Krasnoyarsk, Russia)

Abstract

The paper solves the problem of constructing step adjustment algorithms for a gradient method based on the principle of the steepest descent. The expansion of the step adjustment principle, its formalization and parameterization led the researchers to gradient-type methods with incomplete relaxation or over-relaxation. Such methods require only the gradient of the function to be calculated at the iteration. Optimization of the parameters of the step adaptation algorithms enables us to obtain methods that significantly exceed the steepest descent method in terms of convergence rate. In this paper, we present a universal step adjustment algorithm that does not require selecting optimal parameters. The algorithm is based on orthogonality of successive gradients and replacing complete relaxation with some degree of incomplete relaxation or over-relaxation. Its convergence rate corresponds to algorithms with optimization of the step adaptation algorithm parameters. In our experiments, on average, the proposed algorithm outperforms the steepest descent method by 2.7 times in the number of iterations. The advantage of the proposed methods is their operability under interference conditions. Our paper presents examples of solving test problems in which the interference values are uniformly distributed vectors in a ball with a radius 8 times greater than the gradient norm.

Suggested Citation

  • Vladimir Krutikov & Elena Tovbis & Svetlana Gutova & Ivan Rozhnov & Lev Kazakovtsev, 2024. "Gradient Method with Step Adaptation," Mathematics, MDPI, vol. 13(1), pages 1-35, December.
    • Handle: RePEc:gam:jmathe:v:13:y:2024:i:1:p:61-:d:1554807
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    References listed on IDEAS

    as
    1. NESTEROV, Yurii, 2015. "Universal gradient methods for convex optimization problems," LIDAM Reprints CORE 2701, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
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