# American Institute of Mathematical Sciences

2013, 3(2): 271-282. doi: 10.3934/naco.2013.3.271

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

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

Received  February 2012 Revised  January 2013 Published  April 2013

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.
Citation: Daniel Morales-Silva, David Yang Gao. Complete solutions and triality theory to a nonconvex optimization problem with double-well potential in $\mathbb{R}^n$. Numerical Algebra, Control & Optimization, 2013, 3 (2) : 271-282. doi: 10.3934/naco.2013.3.271
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