# 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

### Fingerprint

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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volume = "naXys-05-2015",
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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

AU - Birgin, Ernesto

AU - Gardenghi, John

AU - Martinez, José-Mario

AU - Santos, Sandra Augusta

AU - Toint, Philippe

PY - 2015/6

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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 $p=2$.

AB - 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 $p=2$.

KW - Nonlinear optimization

KW - Complexity theory

KW - High-order models

M3 - Working paper

VL - naXys-05-2015

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

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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).