Improved worst-case evaluation complexity for potentially rank-deficient nonlinear least-Euclidean-norm problems using higher-order regularized models

Coralia Cartis, Nicholas I M Gould, Philippe Toint

Research output: Working paper

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Abstract

Given a sufficiently smooth vector-valued function $r(x)$,
a local minimizer of $\|r(x)\|_2$ within a closed, non-empty, convex set
$\calF$ is sought by modelling $\|r(x)\|^q_2 / q$
with a $p$-th order Taylor-series approximation plus a $(p+1)$-st order
regularization term for given even $p$ and some appropriate associated $q$.
The resulting algorithm is guaranteed to find a value $\bar{x}$ for which
$\|r(\bar{x})\|_2 \leq \epsilon_p$ or $\chi(\bar{x}) \leq \epsilon_d$, for
some first-order criticality measure $\chi(x)$ of $\|r(x)\|_2$ within $\calF$,
using at most $O(\max\{\max(\epsilon_d,\chi_{\min})^{-(p+1)/p},
\max(\epsilon_p,r_{\min})^{-1/2^i}\})$
evaluations of $r(x)$ and its derivatives;
here $r_{\min}$ and $\chi_{\min} \geq 0$
are any lower bounds on $\|r(x)\|_2$ and $\chi(x)$, respectively,
and $2^i$ is the highest power of $2$ that divides $p$.
Original languageEnglish
Number of pages18
Volume12-2015
Publication statusPublished - 18 Nov 2015

Publication series

NamenaXys technical report

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  • Projects

    Complexity in nonlinear optimization

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

    1/11/08 → …

    Project: Research

    Activities

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

    Philippe Toint (Speaker)

    24 Oct 2017

    Activity: Talk or presentation typesInvited talk

    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

    International Conference on Numerical Analysis and Optimization

    Philippe Toint (Contributor)

    3 Aug 20167 Aug 2016

    Activity: Participating in or organising an event typesParticipation in conference

    Prizes

    Leverhulme Fellow

    TOINT, Philippe (Recipient), Sep 2015

    Prize: Fellowship awarded competitively

    Oliver Smithies Fellow

    TOINT, Philippe (Recipient), Sep 2015

    Prize: Fellowship awarded competitively

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