### Abstract

The complexity of finding e-Approximate first-order critical points for the general smooth constrained optimization problem is shown to be no worse that O(e-2) in terms of function and constraints evaluations. This result is obtained by analyzing the worst-case behaviour of a first-order short-step homotopy algorithm consisting of a feasibility phase followed by an optimization phase, and requires minimal assumptions on the objective function. Since a bound of the same order is known to be valid for the unconstrained case, this leads to the conclusion that the presence of possibly nonlinear/nonconvex inequality/equality constraints is irrelevant for this bound to apply. © 2012 Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society.

Original language | English |
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Pages (from-to) | 93-106 |

Number of pages | 14 |

Journal | Mathematical Programming |

Volume | 144 |

Issue number | 1-2 |

DOIs | |

Publication status | Published - 2014 |

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### Keywords

- Constrained nonlinear optimization
- Evaluation complexity
- Worst-case analysis

### Cite this

*Mathematical Programming*,

*144*(1-2), 93-106. https://doi.org/10.1007/s10107-012-0617-9

}

*Mathematical Programming*, vol. 144, no. 1-2, pp. 93-106. https://doi.org/10.1007/s10107-012-0617-9

**On the complexity of finding first-order critical points in constrained nonlinear optimization.** / Cartis, Coralia; Gould, Nicholas I M; Toint, Philippe L.

Research output: Contribution to journal › Article

TY - JOUR

T1 - On the complexity of finding first-order critical points in constrained nonlinear optimization

AU - Cartis, Coralia

AU - Gould, Nicholas I M

AU - Toint, Philippe L.

N1 - Publication code : FP SB092/2011/13 ; SB04977/2011/13

PY - 2014

Y1 - 2014

N2 - The complexity of finding e-Approximate first-order critical points for the general smooth constrained optimization problem is shown to be no worse that O(e-2) in terms of function and constraints evaluations. This result is obtained by analyzing the worst-case behaviour of a first-order short-step homotopy algorithm consisting of a feasibility phase followed by an optimization phase, and requires minimal assumptions on the objective function. Since a bound of the same order is known to be valid for the unconstrained case, this leads to the conclusion that the presence of possibly nonlinear/nonconvex inequality/equality constraints is irrelevant for this bound to apply. © 2012 Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society.

AB - The complexity of finding e-Approximate first-order critical points for the general smooth constrained optimization problem is shown to be no worse that O(e-2) in terms of function and constraints evaluations. This result is obtained by analyzing the worst-case behaviour of a first-order short-step homotopy algorithm consisting of a feasibility phase followed by an optimization phase, and requires minimal assumptions on the objective function. Since a bound of the same order is known to be valid for the unconstrained case, this leads to the conclusion that the presence of possibly nonlinear/nonconvex inequality/equality constraints is irrelevant for this bound to apply. © 2012 Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society.

KW - Constrained nonlinear optimization

KW - Evaluation complexity

KW - Worst-case analysis

UR - http://www.scopus.com/inward/record.url?scp=84897112534&partnerID=8YFLogxK

U2 - 10.1007/s10107-012-0617-9

DO - 10.1007/s10107-012-0617-9

M3 - Article

VL - 144

SP - 93

EP - 106

JO - Mathematical Programming

JF - Mathematical Programming

SN - 0025-5610

IS - 1-2

ER -