Worst-case evaluation complexity for unconstrained nonlinear optimization using high-order regularized models

Ernesto Birgin; John Gardenghi; José-Mario Martinez; Sandra Augusta Santos; Philippe Toint

Worst-case evaluation complexity for unconstrained nonlinear optimization using high-order regularized models

Ernesto Birgin, John Gardenghi, José-Mario Martinez, Sandra Augusta Santos, Philippe Toint

Namur Institute for Complex Systems

Research output: Working paper

103 Downloads (Pure)

Abstract

The worst-case evaluation complexity for smooth (possibly nonconvex) unconstrained optimization is considered. It is shown that, if one is willing to use derivatives of the objective function up to order $p$ (for $p\geq 1$) and to assume Lipschitz continuity of the $p$-th derivative, then an $\epsilon$-approximate first-order critical point can be computed in at most $O(\epsilon^{-(p+1)/p})$ evaluations of the problem's objective function and its derivatives. This generalizes and subsumes results known for $p=1$ and $p=2$.

Original language	English
Place of Publication	2015
Publisher	Namur center for complex systems
Number of pages	8
Volume	naXys-05-2015
Publication status	Published - Jun 2015

Publication series

Name	naXys Technical Reports
Publisher	naXys
Volume	05-2015

Keywords

Nonlinear optimization
Complexity theory
High-order models

Access to Document

NTR-05-2015Submitted manuscript, 872 KB

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Projects

2 Active

Complexity in nonlinear optimization

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

1/11/08 → …

Project: Research

ADALGOPT: ADALGOPT - Advanced algorithms in nonlinear optimization

SARTENAER, A. & TOINT, P.

1/01/87 → …

Project: Research Axis

Activities

4 Invited talk
4 Oral presentation
1 Research/Teaching in a external institution
1 Visiting 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 types › Invited talk

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

Philippe Toint (Speaker)

24 Oct 2017

Activity: Talk or presentation types › Invited talk

High-order optimality conditions in nonlinear optimization: necessary conditions and a conceptual approach of evaluation complexity

Philippe Toint (Speaker)

10 Aug 2016

Activity: Talk or presentation types › Invited talk

Cite this

Birgin, E., Gardenghi, J., Martinez, J-M., Santos, S. A., & Toint, P. (2015). Worst-case evaluation complexity for unconstrained nonlinear optimization using high-order regularized models. (naXys Technical Reports; Vol. 05-2015). Namur center for complex systems.

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KW - Complexity theory

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