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

Coralia Cartis, N. I. M. Gould, Ph Toint

Research output: Working paper

### Abstract

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}.
Original language English Namur center for complex systems 9 03-2013 Published - 5 May 2013

### Fingerprint

Unconstrained Minimization
Steepest Descent Method
Evaluation Function
Terminate
Iterate
Smooth function
Newton Methods
Lipschitz
Critical point
Objective function
Verify
First-order
Iteration
Path

### Keywords

• Complexity theory, nonlinear optimization, Newton method

### Cite this

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abstract = "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}.",
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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.

Namur center for complex systems, 2013.

Research output: Working paper

TY - UNPB

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

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AU - Gould, N. I. M.

AU - Toint, Ph

PY - 2013/5/5

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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

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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 -