# American Institute of Mathematical Sciences

January  2012, 8(1): 229-242. doi: 10.3934/jimo.2012.8.229

## On the triality theory for a quartic polynomial optimization problem

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

Received  July 2011 Revised  September 2011 Published  November 2011

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.
Citation: David Yang Gao, Changzhi Wu. On the triality theory for a quartic polynomial optimization problem. Journal of Industrial & Management Optimization, 2012, 8 (1) : 229-242. doi: 10.3934/jimo.2012.8.229
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