# 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

Research output: Working paper

### 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 2015 Namur center for complex systems 8 naXys-05-2015 Published - Jun 2015

### Publication series

Name naXys Technical Reports naXys 05-2015

### Keywords

• Nonlinear optimization
• Complexity theory
• High-order models

## Complexity in nonlinear optimization

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

1/11/08 → …

Project: Research

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 typesInvited talk

## 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

## 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 typesInvited 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.