Rates of superlinear convergence for classical quasi-Newton methods

Anton, Rodomanov () and Yurii, Nesterov ()
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Anton, Rodomanov: Université catholique de Louvain, CORE, Belgium
Yurii, Nesterov: Université catholique de Louvain, CORE, Belgium

No 2020011, LIDAM Discussion Papers CORE from Université catholique de Louvain, Center for Operations Research and Econometrics (CORE)

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 $(\frac{nL^2}{μ^2k})^{k/2}$ and $(\frac {nL}{μ k})^{k/2)$ 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.

Keywords: quasi-Newton methods; convex Broyden class; DFP; BFGS; superlinear convergence; local convergence; rate of convergence (search for similar items in EconPapers)
Date: 2020-02-11
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