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An Approach for Analyzing the Global Rate of Convergence of Quasi-Newton and Truncated-Newton Methods

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  • T. L. Jensen

    (Aalborg University)

  • M. Diehl

    (University of Freiburg)

Abstract

Quasi-Newton and truncated-Newton methods are popular methods in optimization and are traditionally seen as useful alternatives to the gradient and Newton methods. Throughout the literature, results are found that link quasi-Newton methods to certain first-order methods under various assumptions. We offer a simple proof to show that a range of quasi-Newton methods are first-order methods in the definition of Nesterov. Further, we define a class of generalized first-order methods and show that the truncated-Newton method is a generalized first-order method and that first-order methods and generalized first-order methods share the same worst-case convergence rates. Further, we extend the complexity analysis for smooth strongly convex problems to finite dimensions. An implication of these results is that in a worst-case scenario, the local superlinear or faster convergence rates of quasi-Newton and truncated-Newton methods cannot be effective unless the number of iterations exceeds half the size of the problem dimension.

Suggested Citation

  • T. L. Jensen & M. Diehl, 2017. "An Approach for Analyzing the Global Rate of Convergence of Quasi-Newton and Truncated-Newton Methods," Journal of Optimization Theory and Applications, Springer, vol. 172(1), pages 206-221, January.
    • Handle: RePEc:spr:joptap:v:172:y:2017:i:1:d:10.1007_s10957-016-1013-z
    DOI: 10.1007/s10957-016-1013-z
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    References listed on IDEAS

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    1. NESTEROV, Yu., 2007. "Gradient methods for minimizing composite objective function," LIDAM Discussion Papers CORE 2007076, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    2. David F. Shanno, 1978. "Conjugate Gradient Methods with Inexact Searches," Mathematics of Operations Research, INFORMS, vol. 3(3), pages 244-256, August.
    3. D.G. Hull, 2002. "On the Huang Class of Variable Metric Methods," Journal of Optimization Theory and Applications, Springer, vol. 113(1), pages 1-4, April.
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