# 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
##### References:
 [1] V. I. Arnold, "Mathematical Methods of Classical Mechanics,", 2nd edition, (1989). doi: 10.1007/978-1-4757-2063-1. [2] C. A. Desoer and B. H. Whalen, A note on pseudoinverses,, Journal of the Society for Industrial and Applied Mathematics, 11 (1963), 442. doi: 10.1137/0111031. [3] D. Y. Gao, "Duality Principles in Nonconvex Systems. Theory Methods and Applications,", Kluwer Academic Publishers, (2000). doi: 10.1007/978-1-4757-3176-7. [4] D. Y. Gao, Perfect duality theory and complete solutions to a class of global optimization problems,, Optim., 52 (2003), 467. doi: 10.1080/02331930310001611501. [5] D. Y. Gao, Nonconvex semi-linear problems and canonical duality solutions,, Advances in Mechanics and Mathematics (eds. D. Y. Gao and R. W. Ogden), (2003), 261. [6] D. Y. Gao, Solutions and optimality to box constrained nonconvex minimization problems,, J. Ind. Manag. Optim., 3 (2007), 293. doi: 10.3934/jimo.2007.3.293. [7] D. Y. Gao, Canonical duality theory: theory, method, and applications in global optimization,, Comput. Chem., 33 (2009), 1964. doi: 10.1016/j.compchemeng.2009.06.009. [8] D. Y. Gao and R. W. Ogden, Multiple solutions to non-convex variational problems with implications for phase transitions and numerical computation,, Quart. J. Mech. Appl. Math., 61 (2008), 497. doi: 10.1093/qjmam/hbn014. [9] D. Y. Gao and N. Ruan, Solutions to quadratic minimization problems with box and integer constraints,, J. Glob. Optim., 47 (2010), 463. doi: 10.1007/s10898-009-9469-0. [10] D. Y. Gao and G. Strang, Geometric nonlinearity: Potential energy, complementary energy, and the gap function,, Quart. Appl. Math., 47 (1989), 487. [11] D. Y. Gao and C. Z. Wu, On the triality theory in global optimization,, J. Industrial and Manegement Optimization, 8 (2012), 229. [12] D. Y. Gao and C. Z. Wu, Triality theory for general unconstrained global optimization problems,, To appear in J. Global Optimization., (). [13] , Maxima, a Computer Algebra System,, Version 5.22.1, (2010). [14] D. M. Morales-Silva and D. Y. Gao, Canonical duality theory and triality for solving general nonconstrained global optimization problems,, To be submitted., (). [15] G. Peters and J. H. Wilkinson, The least squares problem and pseudo-inverses,, The Computer Journal, 13 (1970), 309. doi: 10.1093/comjnl/13.3.309. [16] N. Ruan, D. Y. Gao and Y. Jiao, Canonical dual least square method for solving general nonlinear systems of quadratic equations,, Comput Optim Appl, 47 (2010), 335. doi: 10.1007/s10589-008-9222-5. [17] M. J. Sewell, "Maximum and Minimum Principles,", Cambridge University Press, (1987). [18] Z. B. Wang, S. C. Fang, D. Y. Gao and W. X. Xing, Canonical dual approach to solving the maximum cut problem,, J. Global Optimization, 54 (2012), 341. doi: 10.1007/s10898-012-9881-8. [19] C. Wu, C. J. Li and D. Y. Gao, Canonical primal-dual method for solving non-convex minimization problems,, arXiv:1212.6492, (). [20] R. K. P. Zia, E. F. Redish and S. R. McKay, Making sense of the Legendre transform,, American Journal of Physics, 77 (2009), 614. doi: 10.1119/1.3119512.

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##### References:
 [1] V. I. Arnold, "Mathematical Methods of Classical Mechanics,", 2nd edition, (1989). doi: 10.1007/978-1-4757-2063-1. [2] C. A. Desoer and B. H. Whalen, A note on pseudoinverses,, Journal of the Society for Industrial and Applied Mathematics, 11 (1963), 442. doi: 10.1137/0111031. [3] D. Y. Gao, "Duality Principles in Nonconvex Systems. Theory Methods and Applications,", Kluwer Academic Publishers, (2000). doi: 10.1007/978-1-4757-3176-7. [4] D. Y. Gao, Perfect duality theory and complete solutions to a class of global optimization problems,, Optim., 52 (2003), 467. doi: 10.1080/02331930310001611501. [5] D. Y. Gao, Nonconvex semi-linear problems and canonical duality solutions,, Advances in Mechanics and Mathematics (eds. D. Y. Gao and R. W. Ogden), (2003), 261. [6] D. Y. Gao, Solutions and optimality to box constrained nonconvex minimization problems,, J. Ind. Manag. Optim., 3 (2007), 293. doi: 10.3934/jimo.2007.3.293. [7] D. Y. Gao, Canonical duality theory: theory, method, and applications in global optimization,, Comput. Chem., 33 (2009), 1964. doi: 10.1016/j.compchemeng.2009.06.009. [8] D. Y. Gao and R. W. Ogden, Multiple solutions to non-convex variational problems with implications for phase transitions and numerical computation,, Quart. J. Mech. Appl. Math., 61 (2008), 497. doi: 10.1093/qjmam/hbn014. [9] D. Y. Gao and N. Ruan, Solutions to quadratic minimization problems with box and integer constraints,, J. Glob. Optim., 47 (2010), 463. doi: 10.1007/s10898-009-9469-0. [10] D. Y. Gao and G. Strang, Geometric nonlinearity: Potential energy, complementary energy, and the gap function,, Quart. Appl. Math., 47 (1989), 487. [11] D. Y. Gao and C. Z. Wu, On the triality theory in global optimization,, J. Industrial and Manegement Optimization, 8 (2012), 229. [12] D. Y. Gao and C. Z. Wu, Triality theory for general unconstrained global optimization problems,, To appear in J. Global Optimization., (). [13] , Maxima, a Computer Algebra System,, Version 5.22.1, (2010). [14] D. M. Morales-Silva and D. Y. Gao, Canonical duality theory and triality for solving general nonconstrained global optimization problems,, To be submitted., (). [15] G. Peters and J. H. Wilkinson, The least squares problem and pseudo-inverses,, The Computer Journal, 13 (1970), 309. doi: 10.1093/comjnl/13.3.309. [16] N. Ruan, D. Y. Gao and Y. Jiao, Canonical dual least square method for solving general nonlinear systems of quadratic equations,, Comput Optim Appl, 47 (2010), 335. doi: 10.1007/s10589-008-9222-5. [17] M. J. Sewell, "Maximum and Minimum Principles,", Cambridge University Press, (1987). [18] Z. B. Wang, S. C. Fang, D. Y. Gao and W. X. Xing, Canonical dual approach to solving the maximum cut problem,, J. Global Optimization, 54 (2012), 341. doi: 10.1007/s10898-012-9881-8. [19] C. Wu, C. J. Li and D. Y. Gao, Canonical primal-dual method for solving non-convex minimization problems,, arXiv:1212.6492, (). [20] R. K. P. Zia, E. F. Redish and S. R. McKay, Making sense of the Legendre transform,, American Journal of Physics, 77 (2009), 614. doi: 10.1119/1.3119512.
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