On the evaluation complexity of cubic regularization methods for potentially rank-deficient nonlinear least-squares problems and its relevance to constrained nonlinear optimization

Coralia Cartis, Nick Gould, Philippe Toint

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Abstract

We propose a new termination criterion suitable for potentially singular, zero or nonzero residual, least-squares problems, with which cubic regularization variants take at most O(ε-3/2) residual- and Jacobian-evaluations to drive either the Euclidean norm of the residual or its gradient belowε this is the best known bound for potentially rank-deficient nonlinear least-squares problems. We then apply the new optimality measure and cubic regularization steps to a family of least-squares merit functions in the context of a target-following algorithm for nonlinear equality-constrained problems; this approach yields the first evaluation complexity bound of order ε-3/2 for nonconvexly constrained problems when higher accuracy is required for primal feasibility than for dual first-order criticality.

Original languageEnglish
Pages (from-to)1553-1574
Number of pages22
JournalSIAM Journal on Optimization
Volume23
Issue number3
DOIs
Publication statusPublished - 29 Oct 2013

Keywords

  • Constrained nonlinear optimization
  • Cubic regularization methods
  • Evaluation complexity
  • Least-squares problems
  • Worst-case analysis

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  • Projects

    Complexity in nonlinear optimization

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

    1/11/08 → …

    Project: Research

    Activities

    How much patience do you have? Issues in complexity for nonlinear optimization

    Philippe Toint (Invited speaker)

    5 Feb 2016

    Activity: Talk or presentation typesOral presentation

    How much patience do you have? Issues in complexity for nonlinear optimization

    Philippe Toint (Speaker)

    31 Jan 2016

    Activity: Talk or presentation typesOral presentation

    Polytechnic University of Hong Kong

    Philippe Toint (Visiting researcher)

    31 Jan 201614 Feb 2016

    Activity: Visiting an external institution typesResearch/Teaching in a external institution

    Prizes

    Leverhulme Fellow

    TOINT, Philippe (Recipient), Sep 2015

    Prize: Fellowship awarded competitively

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