Complete solutions and triality theory  to  a nonconvex optimization  problem   with double-well    potential in $\mathbb{R}^n $

Daniel Morales-Silva; David Yang Gao

doi:10.3934/naco.2013.3.271

Article Contents

2013, Volume 3, Issue 2: 271-282. Doi: 10.3934/naco.2013.3.271

This issue Previous Article The stationary iterations revisited Next Article A structured trust region method for nonconvex programming with separable structure

Complete solutions and triality theory to a nonconvex optimization problem with double-well potential in $\mathbb{R}^n $

Daniel Morales-Silva^1, and
David Yang Gao^1,

1.
School of Science, Information Technology and Engineering, University of Ballarat, Mount Helen, VIC 3350

Received: February 2012

Revised: January 2013

Published: April 2013

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Abstract

The main purpose of this research note is to show that the triality theory can always be used to identify both global minimizer and the biggest local maximizer in global optimization. An open problem left on the double-min duality is solved for a nonconvex optimization problem with double-well potential in $\mathbb{R}^n $, which leads to a complete set of analytical solutions. Also a convergency theorem is proved for linear perturbation canonical dual method, which can be used for solving global optimization problems with multiple solutions. The methods and results presented in this note pave the way towards the proof of the triality theory in general cases.

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

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