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

Research output: Working paper

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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

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Unconstrained Optimization

Nonlinear Optimization

Higher Order

Derivatives

Derivative

Evaluation

Objective function

Lipschitz Continuity

Nonconvex Optimization

Critical point

Model

First-order

Generalise

Keywords

Nonlinear optimization
Complexity theory
High-order models

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). 2015: Namur center for complex systems.

Birgin, Ernesto ; Gardenghi, John ; Martinez, José-Mario ; Santos, Sandra Augusta ; Toint, Philippe. / Worst-case evaluation complexity for unconstrained nonlinear optimization using high-order regularized models. 2015 : Namur center for complex systems, 2015. (naXys Technical Reports).

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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 $p=2$.",

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author = "Ernesto Birgin and John Gardenghi and Jos{\'e}-Mario Martinez and Santos, {Sandra Augusta} and Philippe Toint",

year = "2015",

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language = "English",

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Birgin, E, Gardenghi, J, Martinez, J-M, Santos, SA & 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, 2015.

Worst-case evaluation complexity for unconstrained nonlinear optimization using high-order regularized models. / Birgin, Ernesto; Gardenghi, John; Martinez, José-Mario; Santos, Sandra Augusta; Toint, Philippe.

2015 : Namur center for complex systems, 2015. (naXys Technical Reports; Vol. 05-2015).

Research output: Working paper

TY - UNPB

T1 - Worst-case evaluation complexity for unconstrained nonlinear optimization using high-order regularized models

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AU - Gardenghi, John

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AU - Santos, Sandra Augusta

AU - Toint, Philippe

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N2 - 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 $p=2$.

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KW - Complexity theory

KW - High-order models

M3 - Working paper

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PB - Namur center for complex systems

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Birgin E, Gardenghi J, Martinez J-M, Santos SA, Toint P. Worst-case evaluation complexity for unconstrained nonlinear optimization using high-order regularized models. 2015: Namur center for complex systems. 2015 Jun. (naXys Technical Reports).