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Convergence of Newton's method for convex best interpolation

Convergence of Newton's method for convex best interpolation
Convergence of Newton's method for convex best interpolation
In this paper, we consider the problem of finding a convex function which interpolates given points and has a minimal L2 norm of the second derivative. This problem reduces to a system of equations involving semismooth functions. We study a Newton-type method utilizing Clarke's generalized Jacobian and prove that its local convergence is superlinear. For a special choice of a matrix in the generalized Jacobian, we obtain the Newton method proposed by Irvine et al. and settle the question of its convergence. By using a line search strategy, we present a global extension of the Newton method considered. The efficiency of the proposed global strategy is confirmed with numerical experiments.
435-456
Dontchev, Asen L.
98d2f989-952e-4c49-a777-2f43cf5ba132
Qi, Houduo
e9789eb9-c2bc-4b63-9acb-c7e753cc9a85
Qi, Liqun
69936be7-f1aa-4c1f-b403-5bd5f3ba7d4c
Dontchev, Asen L.
98d2f989-952e-4c49-a777-2f43cf5ba132
Qi, Houduo
e9789eb9-c2bc-4b63-9acb-c7e753cc9a85
Qi, Liqun
69936be7-f1aa-4c1f-b403-5bd5f3ba7d4c

Dontchev, Asen L., Qi, Houduo and Qi, Liqun (2001) Convergence of Newton's method for convex best interpolation. Numerische Mathematik, 87 (3), 435-456. (doi:10.1007/PL00005419).

Record type: Article

Abstract

In this paper, we consider the problem of finding a convex function which interpolates given points and has a minimal L2 norm of the second derivative. This problem reduces to a system of equations involving semismooth functions. We study a Newton-type method utilizing Clarke's generalized Jacobian and prove that its local convergence is superlinear. For a special choice of a matrix in the generalized Jacobian, we obtain the Newton method proposed by Irvine et al. and settle the question of its convergence. By using a line search strategy, we present a global extension of the Newton method considered. The efficiency of the proposed global strategy is confirmed with numerical experiments.

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More information

Published date: January 2001
Organisations: Operational Research

Identifiers

Local EPrints ID: 29638
URI: https://eprints.soton.ac.uk/id/eprint/29638
PURE UUID: 0e7c721e-7fe6-40d9-8243-d19b5fd482bf
ORCID for Houduo Qi: ORCID iD orcid.org/0000-0003-3481-4814

Catalogue record

Date deposited: 11 May 2006
Last modified: 07 Aug 2019 00:42

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Contributors

Author: Asen L. Dontchev
Author: Houduo Qi ORCID iD
Author: Liqun Qi

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