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On the Convergence Rate of Quasi-Newton Methods on Strongly Convex Functions with Lipschitz Gradient

Author

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  • Vladimir Krutikov

    (Laboratory “Hybrid Methods of Modeling and Optimization in Complex Systems”, Siberian Federal University, 79 Svobodny Prospekt, Krasnoyarsk 660041, Russia
    Department of Applied Mathematics, Kemerovo State University, 6 Krasnaya Street, Kemerovo 650043, Russia)

  • Elena Tovbis

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

  • Predrag Stanimirović

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

  • Lev Kazakovtsev

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

Abstract

The main results of the study of the convergence rate of quasi-Newton minimization methods were obtained under the assumption that the method operates in the region of the extremum of the function, where there is a stable quadratic representation of the function. Methods based on the quadratic model of the function in the extremum area show significant advantages over classical gradient methods. When solving a specific problem using the quasi-Newton method, a huge number of iterations occur outside the extremum area, unless there is a stable quadratic approximation of the function. In this paper, we study the convergence rate of quasi-Newton-type methods on strongly convex functions with a Lipschitz gradient, without using local quadratic approximations of a function based on the properties of its Hessian. We proved that quasi-Newton methods converge on strongly convex functions with a Lipschitz gradient with the rate of a geometric progression, while the estimate of the convergence rate improves with the increasing number of iterations, which reflects the fact that the learning (adaptation) effect accumulates as the method operates. Another important fact discovered during the theoretical study is the ability of quasi-Newton methods to eliminate the background that slows down the convergence rate. This elimination is achieved through a certain linear transformation that normalizes the elongation of function level surfaces in different directions. All studies were carried out without any assumptions regarding the matrix of second derivatives of the function being minimized.

Suggested Citation

  • Vladimir Krutikov & Elena Tovbis & Predrag Stanimirović & Lev Kazakovtsev, 2023. "On the Convergence Rate of Quasi-Newton Methods on Strongly Convex Functions with Lipschitz Gradient," Mathematics, MDPI, vol. 11(23), pages 1-15, November.
  • Handle: RePEc:gam:jmathe:v:11:y:2023:i:23:p:4715-:d:1284752
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    References listed on IDEAS

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    1. Anton Rodomanov & Yurii Nesterov, 2021. "New Results on Superlinear Convergence of Classical Quasi-Newton Methods," Journal of Optimization Theory and Applications, Springer, vol. 188(3), pages 744-769, March.
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