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A Family of Multi-Step Subgradient Minimization Methods

Author

Listed:
  • Elena Tovbis

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

  • Vladimir Krutikov

    (Department of Applied Mathematics, Kemerovo State University, 6 Krasnaya Street, Kemerovo 650043, Russia
    Faculty of Sciences and Mathematics, University of Nis, 18000 Nis, Serbia)

  • Predrag Stanimirović

    (Faculty of Sciences and Mathematics, University of Nis, 18000 Nis, Serbia
    Laboratory “Hybrid Methods of Modeling and Optimization in Complex Systems”, Siberian Federal University, 79 Svobodny Prospekt, Krasnoyarsk 660041, Russia)

  • Vladimir Meshechkin

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

  • Aleksey Popov

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

  • Lev Kazakovtsev

    (Institute of Informatics and Telecommunications, Reshetnev Siberian State University of Science and Technology, 31 Krasnoyarskii Rabochii Prospekt, Krasnoyarsk 660037, Russia
    Faculty of Sciences and Mathematics, University of Nis, 18000 Nis, Serbia)

Abstract

For solving non-smooth multidimensional optimization problems, we present a family of relaxation subgradient methods (RSMs) with a built-in algorithm for finding the descent direction that forms an acute angle with all subgradients in the neighborhood of the current minimum. Minimizing the function along the opposite direction (with a minus sign) enables the algorithm to go beyond the neighborhood of the current minimum. The family of algorithms for finding the descent direction is based on solving systems of inequalities. The finite convergence of the algorithms on separable bounded sets is proved. Algorithms for solving systems of inequalities are used to organize the RSM family. On quadratic functions, the methods of the RSM family are equivalent to the conjugate gradient method (CGM). The methods are intended for solving high-dimensional problems and are studied theoretically and numerically. Examples of solving convex and non-convex smooth and non-smooth problems of large dimensions are given.

Suggested Citation

  • Elena Tovbis & Vladimir Krutikov & Predrag Stanimirović & Vladimir Meshechkin & Aleksey Popov & Lev Kazakovtsev, 2023. "A Family of Multi-Step Subgradient Minimization Methods," Mathematics, MDPI, vol. 11(10), pages 1-24, May.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:10:p:2264-:d:1145064
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    References listed on IDEAS

    as
    1. Igor Konnov, 2020. "A Non-monotone Conjugate Subgradient Type Method for Minimization of Convex Functions," Journal of Optimization Theory and Applications, Springer, vol. 184(2), pages 534-546, February.
    2. Yutao Zheng & Bing Zheng, 2017. "Two New Dai–Liao-Type Conjugate Gradient Methods for Unconstrained Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 175(2), pages 502-509, November.
    3. M. V. Solodov & S. K. Zavriev, 1998. "Error Stability Properties of Generalized Gradient-Type Algorithms," Journal of Optimization Theory and Applications, Springer, vol. 98(3), pages 663-680, September.
    4. Junyu Lu & Yong Li & Hongtruong Pham, 2020. "A Modified Dai–Liao Conjugate Gradient Method with a New Parameter for Solving Image Restoration Problems," Mathematical Problems in Engineering, Hindawi, vol. 2020, pages 1-13, September.
    5. Auwal Bala Abubakar & Poom Kumam & Hassan Mohammad & Aliyu Muhammed Awwal & Kanokwan Sitthithakerngkiet, 2019. "A Modified Fletcher–Reeves Conjugate Gradient Method for Monotone Nonlinear Equations with Some Applications," Mathematics, MDPI, vol. 7(8), pages 1-25, August.
    6. 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).
    7. Y.H. Dai & Y. Yuan, 2001. "An Efficient Hybrid Conjugate Gradient Method for Unconstrained Optimization," Annals of Operations Research, Springer, vol. 103(1), pages 33-47, March.
    8. Vladimir Krutikov & Svetlana Gutova & Elena Tovbis & Lev Kazakovtsev & Eugene Semenkin, 2022. "Relaxation Subgradient Algorithms with Machine Learning Procedures," Mathematics, MDPI, vol. 10(21), pages 1-33, October.
    Full references (including those not matched with items on IDEAS)

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