On the triality theory for a quartic polynomial optimization problem

David Yang Gao; Changzhi Wu

doi:10.3934/jimo.2012.8.229

Article Contents

2012, Volume 8, Issue 1: 229-242. Doi: 10.3934/jimo.2012.8.229

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On the triality theory for a quartic polynomial optimization problem

David Yang Gao^1, and
Changzhi Wu^2,

1.
School of Sciences, Information Technology and Engineering, University of Ballarat, Victoria 3353
2.
School of Science, Information Technology and Engineering, University of Ballarat, Victoria 3353

Received: June 30, 2011

Revised: August 31, 2011

Published: December 2011

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Abstract

This paper presents a detailed proof of the triality theorem for a class of fourth-order polynomial optimization problems. The method is based on linear algebra but it solves an open problem on the double-min duality. Results show that the triality theory holds strongly in the tri-duality form for our problem if the primal problem and its canonical dual have the same dimension; otherwise, both the canonical min-max duality and the double-max duality still hold strongly, but the double-min duality holds weakly in a symmetrical form. Some numerical examples are presented to illustrate that this theory can be used to identify not only the global minimum, but also the local minimum and local maximum.

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Mathematics Subject Classification: 90C26, 90C30, 90C46, 90C49.

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