Worst-case evaluation complexity and optimality of second-order methods for nonconvex smooth optimization

Coralia Cartis, Nicholas I.M. Gould, Philippe Toint

Research output: Contribution in Book/Catalog/Report/Conference proceedingConference contribution

Abstract

We establish or refute the optimality of inexact second-order methods for unconstrained nonconvex optimization from the point of view of worst-case evaluation complexity, improving and generalizing our previous results. To this aim, we consider a new general class of inexact second-order algorithms for unconstrained optimization that includes regularization and trust-region variations of Newton’s method as well as of their linesearch variants. For each method in this class and arbitrary accuracy threshold ∊ 2 (0; 1), we exhibit a smooth objective function with bounded range, whose gradient is globally Lipschitz continuous and whose Hessian is α-Hölder continuous (for given α 2 [0; 1]), for which the method in question takes at least b∊-(2+α)/(1+α)c function evaluations to generate a first iterate whose gradient is smaller than ∊ in norm. Moreover, we also construct another function on which Newton’s takes b∊-2c evaluations, but whose Hessian is Lipschitz continuous on the path of iterates. These examples provide lower bounds on the worst-case evaluation complexity of methods in our class when applied to smooth problems satisfying the relevant assumptions. Furthermore, for α = 1, this lower bound is of the same order in ∊ as the upper bound on the worst-case evaluation complexity of the cubic regularization method and other algorithms in a class of methods recently proposed by Curtis, Robinson and Samadi or by Royer and Wright, thus implying that these methods have optimal worst-case evaluation complexity within a wider class of second-order methods, and that Newton’s method is suboptimal.

Original languageEnglish
Title of host publicationInvited Lectures
EditorsBoyan Sirakov, Paulo Ney de Souza, Marcelo Viana
PublisherWorld Scientific Publishing Co Pte Ltd
Pages3729-3768
Number of pages40
ISBN (Electronic)9789813272934
Publication statusPublished - 1 Jan 2018
Event2018 International Congress of Mathematicians, ICM 2018 - Rio de Janeiro, Brazil
Duration: 1 Aug 20189 Aug 2018

Publication series

NameProceedings of the International Congress of Mathematicians, ICM 2018
Volume4

Conference

Conference2018 International Congress of Mathematicians, ICM 2018
CountryBrazil
CityRio de Janeiro
Period1/08/189/08/18

Keywords

  • Nonlinear optimization
  • Evaluation complexity
  • Regularization methods

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  • Projects

    Complexity in nonlinear optimization

    TOINT, P., Gould, N. I. M. & Cartis, C.

    1/11/08 → …

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    Cite this

    Cartis, C., Gould, N. I. M., & Toint, P. (2018). Worst-case evaluation complexity and optimality of second-order methods for nonconvex smooth optimization. In B. Sirakov, P. N. de Souza, & M. Viana (Eds.), Invited Lectures (pp. 3729-3768). (Proceedings of the International Congress of Mathematicians, ICM 2018; Vol. 4). World Scientific Publishing Co Pte Ltd.