Research output per year
Research output per year
Coralia Cartis, Nicholas I.M. Gould, Philippe Toint
Research output: Contribution in Book/Catalog/Report/Conference proceeding › Conference contribution
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∊-^{2}c 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 language | English |
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Title of host publication | Invited Lectures |
Editors | Boyan Sirakov, Paulo Ney de Souza, Marcelo Viana |
Publisher | World Scientific Publishing Co Pte Ltd |
Pages | 3729-3768 |
Number of pages | 40 |
ISBN (Electronic) | 9789813272934 |
Publication status | Published - 1 Jan 2018 |
Event | 2018 International Congress of Mathematicians, ICM 2018 - Rio de Janeiro, Brazil Duration: 1 Aug 2018 → 9 Aug 2018 |
Name | Proceedings of the International Congress of Mathematicians, ICM 2018 |
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Volume | 4 |
Conference | 2018 International Congress of Mathematicians, ICM 2018 |
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Country/Territory | Brazil |
City | Rio de Janeiro |
Period | 1/08/18 → 9/08/18 |
Research output: Book/Report/Journal › Book
Research output: Contribution to journal › Article › peer-review
Research output: Contribution to journal › Article › peer-review
TOINT, P., Gould, N. I. M. & Cartis, C.
1/11/08 → …
Project: Research