# Second-order optimality and beyond: characterization and evaluation complexity in nonconvex convexly-constrained optimization

Coralia Cartis, Nicholas I M Gould, Philippe Toint

Résultats de recherche: Contribution à un journal/une revueArticle

### Résumé

High-order optimality conditions for convexly-constrained nonlinear
optimization problems are analyzed. A corresponding (expensive) measure of criticality for arbitrary order is proposed and extended to define high-order $\epsilon$-approximate critical points. This new measure is then used within a conceptual trust-region algorithm to show that, if derivatives of theobjective function up to order $q \geq 1$ can be evaluated and are Lipschitz continuous, then this algorithm applied to the convexly constrained problem needs at most $O(\epsilon^{-(q+1)})$ evaluations of $f$ and its derivatives to compute an $\epsilon$-approximate $q$-th order critical point. This provides the first evaluation complexity result for critical points of arbitrary order in nonlinear optimization. An example is discussed showing that the obtained evaluation complexity bounds are essentially sharp.
langue Anglais Foundations of Computational Mathematics Accepté/sous presse - 28 mai 2017

### Empreinte digitale

Nonconvex Optimization
Constrained Optimization
Optimality
Evaluation
Constrained optimization
Derivatives
Critical point
Nonlinear Optimization
Higher Order
Derivative
Arbitrary
Trust Region Algorithm
Order Conditions
Criticality
Optimality Conditions
Lipschitz
Nonlinear Problem
Optimization Problem

### Citer ceci

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author = "Coralia Cartis and Gould, {Nicholas I M} and Philippe Toint",
year = "2017",
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journal = "Foundations of Computational Mathematics",
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Second-order optimality and beyond : characterization and evaluation complexity in nonconvex convexly-constrained optimization. / Cartis, Coralia; Gould, Nicholas I M; Toint, Philippe.

Dans: Foundations of Computational Mathematics, 28.05.2017.

Résultats de recherche: Contribution à un journal/une revueArticle

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T1 - Second-order optimality and beyond

T2 - Foundations of Computational Mathematics

AU - Cartis,Coralia

AU - Gould,Nicholas I M

AU - Toint,Philippe

PY - 2017/5/28

Y1 - 2017/5/28

N2 - High-order optimality conditions for convexly-constrained nonlinearoptimization problems are analyzed. A corresponding (expensive) measure of criticality for arbitrary order is proposed and extended to define high-order $\epsilon$-approximate critical points. This new measure is then used within a conceptual trust-region algorithm to show that, if derivatives of theobjective function up to order $q \geq 1$ can be evaluated and are Lipschitz continuous, then this algorithm applied to the convexly constrained problem needs at most $O(\epsilon^{-(q+1)})$ evaluations of $f$ and its derivatives to compute an $\epsilon$-approximate $q$-th order critical point. This provides the first evaluation complexity result for critical points of arbitrary order in nonlinear optimization. An example is discussed showing that the obtained evaluation complexity bounds are essentially sharp.

AB - High-order optimality conditions for convexly-constrained nonlinearoptimization problems are analyzed. A corresponding (expensive) measure of criticality for arbitrary order is proposed and extended to define high-order $\epsilon$-approximate critical points. This new measure is then used within a conceptual trust-region algorithm to show that, if derivatives of theobjective function up to order $q \geq 1$ can be evaluated and are Lipschitz continuous, then this algorithm applied to the convexly constrained problem needs at most $O(\epsilon^{-(q+1)})$ evaluations of $f$ and its derivatives to compute an $\epsilon$-approximate $q$-th order critical point. This provides the first evaluation complexity result for critical points of arbitrary order in nonlinear optimization. An example is discussed showing that the obtained evaluation complexity bounds are essentially sharp.

KW - Complexity theory

KW - optimality conditions

KW - nonlinear optimization

M3 - Article

JO - Foundations of Computational Mathematics

JF - Foundations of Computational Mathematics

SN - 1615-3375

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