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Rates of superlinear convergence for classical quasi-Newton methods

Author

Listed:
  • Rodomanov, Anton

    (Université catholique de Louvain, ICTEAM)

  • Nesterov, Yurii

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

Abstract

We study the local convergence of classical quasi-Newton methods for nonlinear optimization. Although it was well established a long time ago that asymptotically these methods converge superlinearly, the corresponding rates of convergence still remain unknown. In this paper, we address this problem. We obtain first explicit non-asymptotic rates of superlinear convergence for the standard quasi-Newton methods, which are based on the updating formulas from the convex Broyden class. In particular, for the well-known DFP and BFGS methods, we obtain the rates of the form (nL2μ2k)k/2 and (nLμk)k/2 respectively, where k is the iteration counter, n is the dimension of the problem, μ is the strong convexity parameter, and L is the Lipschitz constant of the gradient.

Suggested Citation

  • Rodomanov, Anton & Nesterov, Yurii, 2023. "Rates of superlinear convergence for classical quasi-Newton methods," LIDAM Reprints CORE 3238, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    • Handle: RePEc:cor:louvrp:3238
    DOI: https://doi.org/10.1007/s10107-021-01622-5
    Note: In: Mathematical Programming, 2022, vol. 194, p. 159-190
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