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Optimal step length for the Newton method: case of self-concordant functions

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  • Roland Hildebrand

    (Univ. Grenoble Alpes)

Abstract

The theoretical foundation of path-following methods is the performance analysis of the (damped) Newton step on the class of self-concordant functions. However, the bounds available in the literature and used in the design of path-following methods are not optimal. In this contribution we use methods of optimal control theory to compute the optimal step length of the Newton method on the class of self-concordant functions, as a function of the initial Newton decrement, and the resulting worst-case decrease of the decrement. The exact bounds are expressed in terms of solutions of ordinary differential equations which cannot be integrated explicitly. We provide approximate numerical and analytic expressions which are accurate enough for use in optimization methods. Consequently, the neighbourhood of the central path in which the iterates of path-following methods are required to stay can be enlarged, enabling faster progress along the central path during each iteration and hence fewer iterations to achieve a given accuracy.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:mathme:v:94:y:2021:i:2:d:10.1007_s00186-021-00755-9
    DOI: 10.1007/s00186-021-00755-9
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    References listed on IDEAS

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    1. TAYLOR, Adrien B. & HENDRICKX, Julien M. & François GLINEUR, 2016. "Exact worst-case performance of first-order methods for composite convex optimization," LIDAM Discussion Papers CORE 2016052, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    2. Yurii Nesterov, 2018. "Lectures on Convex Optimization," Springer Optimization and Its Applications, Springer, edition 2, number 978-3-319-91578-4, June.
    3. de Klerk, Etienne & Glineur, Francois & Taylor, Adrien, 2020. "Worst-case convergence analysis of inexact gradient and Newton methods through semidefinite programming performance estimation," Other publications TiSEM 03a65c83-e88f-4723-a50e-2, Tilburg University, School of Economics and Management.
    4. Daniel Ralph, 1994. "Global Convergence of Damped Newton's Method for Nonsmooth Equations via the Path Search," Mathematics of Operations Research, INFORMS, vol. 19(2), pages 352-389, May.
    5. DE KLERK, Etienne & GLINEUR, François & TAYLOR, Adrien B., 2016. "On the Worst-case Complexity of the Gradient Method with Exact Line Search for Smooth Strongly Convex Functions," LIDAM Discussion Papers CORE 2016027, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
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    Cited by:

    1. Roland Hildebrand, 2022. "Semi-definite Representations for Sets of Cubics on the Two-dimensional Sphere," Journal of Optimization Theory and Applications, Springer, vol. 195(2), pages 666-675, November.

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