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Worst-Case Convergence Analysis of Inexact Gradient and Newton Methods Through Semidefinite Programming Performance Estimation

Author

Listed:
  • De Klerk, Etienne
  • Glineur, François

    (Université catholique de Louvain, LIDAM/CORE, Belgium)

  • Taylor, Adrien B.

Abstract

We provide new tools for worst-case performance analysis of the gradient (or steepest descent) method of Cauchy for smooth strongly convex functions, and Newton’s method for selfconcordant functions, including the case of inexact search directions. The analysis uses semidefinite programming performance estimation, as pioneered by Drori and Teboulle [Math. Program., 145 (2014), pp. 451–482], and extends recent performance estimation results for the method of Cauchy by the authors [Optim. Lett., 11 (2017), pp. 1185–1199]. To illustrate the applicability of the tools, we demonstrate a novel complexity analysis of short step interior point methods using inexact search directions. As an example in this framework, we sketch how to give a rigorous worst-case complexity analysis of a recent interior point method by Abernethy and and Hazan [PMLR, 48 (2016), pp. 2520– 2528].

Suggested Citation

  • De Klerk, Etienne & Glineur, François & Taylor, Adrien B., 2020. "Worst-Case Convergence Analysis of Inexact Gradient and Newton Methods Through Semidefinite Programming Performance Estimation," LIDAM Reprints CORE 3134, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    • Handle: RePEc:cor:louvrp:3134
    DOI: https://doi.org/10.1137/19m1281368
    Note: In : SIAM Journal on Optimization - Vol. 30, no.3, p. 2053-2082 (2020)
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    Cited by:

    1. Abbaszadehpeivasti, Hadi, 2024. "Performance analysis of optimization methods for machine learning," Other publications TiSEM 3050a62d-1a1f-494e-99ef-7, Tilburg University, School of Economics and Management.
    2. André Uschmajew & Bart Vandereycken, 2022. "A Note on the Optimal Convergence Rate of Descent Methods with Fixed Step Sizes for Smooth Strongly Convex Functions," Journal of Optimization Theory and Applications, Springer, vol. 194(1), pages 364-373, July.
    3. Abbaszadehpeivasti, Hadi & de Klerk, Etienne & Zamani, Moslem, 2023. "Convergence rate analysis of randomized and cyclic coordinate descent for convex optimization through semidefinite programming," Other publications TiSEM 88512ac0-c26a-4a99-b840-3, Tilburg University, School of Economics and Management.
    4. Abbaszadehpeivasti, Hadi & de Klerk, Etienne & Zamani, Moslem, 2022. "The exact worst-case convergence rate of the gradient method with fixed step lengths for L-smooth functions," Other publications TiSEM 061688c6-f97c-4024-bb5b-1, Tilburg University, School of Economics and Management.
    5. Roland Hildebrand, 2021. "Optimal step length for the Newton method: case of self-concordant functions," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 94(2), pages 253-279, October.
    6. Hadi Abbaszadehpeivasti & Etienne Klerk & Moslem Zamani, 2024. "On the Rate of Convergence of the Difference-of-Convex Algorithm (DCA)," Journal of Optimization Theory and Applications, Springer, vol. 202(1), pages 475-496, July.

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