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Armijo Newton method for convex best interpolation

Armijo Newton method for convex best interpolation
Armijo Newton method for convex best interpolation
Newton's method with Armijo line search (Armijo Newton method) has been practically known extremely efficient for the problem of convex best interpolation and numerical experiment strongly indicates its global convergence. However, similar to the classical Newton method, the Newton matrix far from the solution may be singular or near singular, posing a great deal of difficulties in proving the global convergence of Armijo Newton method. By employing the objective function of Lagrange dual problem, it is observed that whenever the Newton matrix is near singular at some point, one can easily [at cost of O(N), N is the dimension of the problem] find a nearby point which has a well-conditioned Newton matrix and lower function value. In this case, Armijo Newton method starts from this nearby point. We prove that this slightly modified Armijo Newton method is globally as well as locally quadratically convergent, and in an important case, it also has finite termination property. Numerical results demonstrate the efficiency of the proposed method.
Newton's method, convex interpolation, line search
1055-6788
179-200
Qi, Hou-Duo
e9789eb9-c2bc-4b63-9acb-c7e753cc9a85
Yang, X.Q.
b0344a02-7c45-4034-ac68-c396e77ac9e5
Qi, Hou-Duo
e9789eb9-c2bc-4b63-9acb-c7e753cc9a85
Yang, X.Q.
b0344a02-7c45-4034-ac68-c396e77ac9e5

Qi, Hou-Duo and Yang, X.Q. (2006) Armijo Newton method for convex best interpolation. Optimization Methods and Software, 21 (2), 179-200. (doi:10.1080/10556780500065432).

Record type: Article

Abstract

Newton's method with Armijo line search (Armijo Newton method) has been practically known extremely efficient for the problem of convex best interpolation and numerical experiment strongly indicates its global convergence. However, similar to the classical Newton method, the Newton matrix far from the solution may be singular or near singular, posing a great deal of difficulties in proving the global convergence of Armijo Newton method. By employing the objective function of Lagrange dual problem, it is observed that whenever the Newton matrix is near singular at some point, one can easily [at cost of O(N), N is the dimension of the problem] find a nearby point which has a well-conditioned Newton matrix and lower function value. In this case, Armijo Newton method starts from this nearby point. We prove that this slightly modified Armijo Newton method is globally as well as locally quadratically convergent, and in an important case, it also has finite termination property. Numerical results demonstrate the efficiency of the proposed method.

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More information

Published date: April 2006
Keywords: Newton's method, convex interpolation, line search
Organisations: Operational Research

Identifiers

Local EPrints ID: 54540
URI: http://eprints.soton.ac.uk/id/eprint/54540
ISSN: 1055-6788
PURE UUID: cef6b137-6efc-4f17-8b9f-93aa34df5a8a
ORCID for Hou-Duo Qi: ORCID iD orcid.org/0000-0003-3481-4814

Catalogue record

Date deposited: 28 Jul 2008
Last modified: 20 Jul 2019 00:57

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