### Abstract

Original language | English |
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Publisher | Namur center for complex systems |

Number of pages | 9 |

Volume | 03-2013 |

Publication status | Published - 5 May 2013 |

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### Keywords

- Complexity theory, nonlinear optimization, Newton method

### Cite this

*An example of slow convergence for Newton's method on a function with globally Lipschitz continuous Hessian*. Namur center for complex systems.

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**An example of slow convergence for Newton's method on a function with globally Lipschitz continuous Hessian.** / Cartis, Coralia; Gould, N. I. M.; Toint, Ph.

Research output: Working paper

TY - UNPB

T1 - An example of slow convergence for Newton's method on a function with globally Lipschitz continuous Hessian

AU - Cartis, Coralia

AU - Gould, N. I. M.

AU - Toint, Ph

PY - 2013/5/5

Y1 - 2013/5/5

N2 - An example is presented where Newton's method for unconstrained minimization is applied to find an $\epsilon$-approximate first-order critical point of a smooth function and takes a multiple of $\epsilon^{-2}$ iterations and function evaluations to terminate, which is as many as the steepest-descent method in its worst-case. The novel feature of the proposed example is that the objective function has a globally Lipschitz-continuous Hessian, while a previous example published by the same authors only ensured this critical property along the path of iterates, which is impossible to verify \emph{a priori}.

AB - An example is presented where Newton's method for unconstrained minimization is applied to find an $\epsilon$-approximate first-order critical point of a smooth function and takes a multiple of $\epsilon^{-2}$ iterations and function evaluations to terminate, which is as many as the steepest-descent method in its worst-case. The novel feature of the proposed example is that the objective function has a globally Lipschitz-continuous Hessian, while a previous example published by the same authors only ensured this critical property along the path of iterates, which is impossible to verify \emph{a priori}.

KW - Complexity theory, nonlinear optimization, Newton method

M3 - Working paper

VL - 03-2013

BT - An example of slow convergence for Newton's method on a function with globally Lipschitz continuous Hessian

PB - Namur center for complex systems

ER -