Complexity of partially separable convexly constrained optimization with non-Lipschitzian singularities

Xiaojun Chen, Philippe Toint, Hong Wang

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Abstract

An adaptive regularization algorithm using high-order models is proposed for solving partially separable convexly constrained nonlinear optimization problems whose objective function contains non-Lipschitzian 'q-norm regularization terms for q ϵ (0; 1). It is shown that the algorithm using a pth-order Taylor model for p odd needs in general at most O(ϵ -(p+1)=p) evaluations of the objective function and its derivatives (at points where they are defined) to produce an ϵ-approximate first-order critical point. This result is obtained either with Taylor models, at the price of requiring the feasible set to be kernel centered" (which includes bound constraints and many other cases of interest), or with non-Lipschitz models, at the price of passing the difficulty to the computation of the step. Since this complexity bound is identical in order to that already known for purely Lipschitzian minimization subject to convex constraints [C. Cartis, N. I. M. Gould, and Ph. L. Toint, IMA J. Numer. Anal., 32 (2012), pp. 1662-1695], the new result shows that introducing non-Lipschitzian singularities in the objective function may not affect the worst-case evaluation complexity order. The result also shows that using the problem's partially separable structure (if present) does not affect the complexity order either. A final (worse) complexity bound is derived for the case where Taylor models are used with a general convex feasible set.

Original languageEnglish
Pages (from-to)874-903
Number of pages30
JournalSIAM Journal on Optimization
Volume29
Issue number1
DOIs
Publication statusPublished - 15 Apr 2019

Keywords

  • Non_Lipschitz optimization
  • Complexity theory
  • partial separability

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  • Projects

    Complexity in nonlinear optimization

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

    1/11/08 → …

    Project: Research

    Activities

    • 2 Invited talk
    • 1 Visiting an external academic institution

    Recent progress in evaluation complexity for nonlinear optimization

    Philippe Toint (Speaker)

    4 Oct 2019

    Activity: Talk or presentation typesInvited talk

    Departement of Applied Mathematics, Polytechnic University of Hong Kong

    Philippe Toint (Visiting researcher)

    15 Nov 201815 Dec 2018

    Activity: Visiting an external institution typesVisiting an external academic institution

    A path and some adventures in the jungle of high-order nonlinear optimization

    Philippe Toint (Speaker)

    23 Oct 2017

    Activity: Talk or presentation typesInvited talk

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