# 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 and 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, Springer-Verlag, Berlin, Heidelberg, 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), 442-447. doi: 10.1137/0111031. [3] D. Y. Gao, "Duality Principles in Nonconvex Systems. Theory Methods and Applications," Kluwer Academic Publishers, Dordrecht/Boston/London, 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), 467-493. 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), Kluwer, (2003), 261-311. [6] D. Y. Gao, Solutions and optimality to box constrained nonconvex minimization problems, J. Ind. Manag. Optim., 3 (2007), 293-304. 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), 1964-1972. 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), 497-522. 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), 463-484. 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), 487-504. [11] D. Y. Gao and C. Z. Wu, On the triality theory in global optimization, J. Industrial and Manegement Optimization, 8 (2012), 229-242. Also published in arXiv:1104.2970v1 at http://arxiv.org/abs/1104.2970. [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), 309-316. 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), 335-347. doi: 10.1007/s10589-008-9222-5. [17] M. J. Sewell, "Maximum and Minimum Principles," Cambridge University Press, Cambridge, New York, Port Chester, Melbourne, Sydney, 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), 341-352. 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), 614-622. doi: 10.1119/1.3119512.

show all references

##### References:
 [1] V. I. Arnold, "Mathematical Methods of Classical Mechanics," 2nd edition, Springer-Verlag, Berlin, Heidelberg, 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), 442-447. doi: 10.1137/0111031. [3] D. Y. Gao, "Duality Principles in Nonconvex Systems. Theory Methods and Applications," Kluwer Academic Publishers, Dordrecht/Boston/London, 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), 467-493. 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), Kluwer, (2003), 261-311. [6] D. Y. Gao, Solutions and optimality to box constrained nonconvex minimization problems, J. Ind. Manag. Optim., 3 (2007), 293-304. 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), 1964-1972. 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), 497-522. 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), 463-484. 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), 487-504. [11] D. Y. Gao and C. Z. Wu, On the triality theory in global optimization, J. Industrial and Manegement Optimization, 8 (2012), 229-242. Also published in arXiv:1104.2970v1 at http://arxiv.org/abs/1104.2970. [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), 309-316. 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), 335-347. doi: 10.1007/s10589-008-9222-5. [17] M. J. Sewell, "Maximum and Minimum Principles," Cambridge University Press, Cambridge, New York, Port Chester, Melbourne, Sydney, 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), 341-352. 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), 614-622. doi: 10.1119/1.3119512.
 [1] David Yang Gao, Changzhi Wu. On the triality theory for a quartic polynomial optimization problem. Journal of Industrial and Management Optimization, 2012, 8 (1) : 229-242. doi: 10.3934/jimo.2012.8.229 [2] Manxue You, Shengjie Li. Perturbation of Image and conjugate duality for vector optimization. Journal of Industrial and Management Optimization, 2022, 18 (2) : 731-745. doi: 10.3934/jimo.2020176 [3] Yong Xia, Ruey-Lin Sheu, Shu-Cherng Fang, Wenxun Xing. Double well potential function and its optimization in the $N$ -dimensional real space-part Ⅱ. Journal of Industrial and Management Optimization, 2017, 13 (3) : 1307-1328. doi: 10.3934/jimo.2016074 [4] Shu-Cherng Fang, David Y. Gao, Gang-Xuan Lin, Ruey-Lin Sheu, Wenxun Xing. Double well potential function and its optimization in the $N$ -dimensional real space-part Ⅰ. Journal of Industrial and Management Optimization, 2017, 13 (3) : 1291-1305. doi: 10.3934/jimo.2016073 [5] Yu Chen, Yanheng Ding, Tian Xu. Potential well and multiplicity of solutions for nonlinear Dirac equations. Communications on Pure and Applied Analysis, 2020, 19 (1) : 587-607. doi: 10.3934/cpaa.2020028 [6] Yacheng Liu, Runzhang Xu. Potential well method for initial boundary value problem of the generalized double dispersion equations. Communications on Pure and Applied Analysis, 2008, 7 (1) : 63-81. doi: 10.3934/cpaa.2008.7.63 [7] Radu Strugariu, Mircea D. Voisei, Constantin Zălinescu. Counter-examples in bi-duality, triality and tri-duality. Discrete and Continuous Dynamical Systems, 2011, 31 (4) : 1453-1468. doi: 10.3934/dcds.2011.31.1453 [8] Carlos F. Daganzo. On the variational theory of traffic flow: well-posedness, duality and applications. Networks and Heterogeneous Media, 2006, 1 (4) : 601-619. doi: 10.3934/nhm.2006.1.601 [9] Thomas Bartsch, Zhongwei Tang. Multibump solutions of nonlinear Schrödinger equations with steep potential well and indefinite potential. Discrete and Continuous Dynamical Systems, 2013, 33 (1) : 7-26. doi: 10.3934/dcds.2013.33.7 [10] Tadahiro Oh, Mamoru Okamoto, Oana Pocovnicu. On the probabilistic well-posedness of the nonlinear Schrödinger equations with non-algebraic nonlinearities. Discrete and Continuous Dynamical Systems, 2019, 39 (6) : 3479-3520. doi: 10.3934/dcds.2019144 [11] John Boyd. Strongly nonlinear perturbation theory for solitary waves and bions. Evolution Equations and Control Theory, 2019, 8 (1) : 1-29. doi: 10.3934/eect.2019001 [12] Yubo Yuan. Canonical duality solution for alternating support vector machine. Journal of Industrial and Management Optimization, 2012, 8 (3) : 611-621. doi: 10.3934/jimo.2012.8.611 [13] Jutamas Kerdkaew, Rabian Wangkeeree, Rattanaporn Wangkeeree. Global optimality conditions and duality theorems for robust optimal solutions of optimization problems with data uncertainty, using underestimators. Numerical Algebra, Control and Optimization, 2022, 12 (1) : 93-107. doi: 10.3934/naco.2021053 [14] Changxing Miao, Bo Zhang. Global well-posedness of the Cauchy problem for nonlinear Schrödinger-type equations. Discrete and Continuous Dynamical Systems, 2007, 17 (1) : 181-200. doi: 10.3934/dcds.2007.17.181 [15] Yonggeun Cho, Gyeongha Hwang, Tohru Ozawa. Global well-posedness of critical nonlinear Schrödinger equations below $L^2$. Discrete and Continuous Dynamical Systems, 2013, 33 (4) : 1389-1405. doi: 10.3934/dcds.2013.33.1389 [16] Markus Musch, Ulrik Skre Fjordholm, Nils Henrik Risebro. Well-posedness theory for nonlinear scalar conservation laws on networks. Networks and Heterogeneous Media, 2022, 17 (1) : 101-128. doi: 10.3934/nhm.2021025 [17] Irena Lasiecka, Roberto Triggiani. Global exact controllability of semilinear wave equations by a double compactness/uniqueness argument. Conference Publications, 2005, 2005 (Special) : 556-565. doi: 10.3934/proc.2005.2005.556 [18] A. Rodríguez-Bernal. Perturbation of the exponential type of linear nonautonomous parabolic equations and applications to nonlinear equations. Discrete and Continuous Dynamical Systems, 2009, 25 (3) : 1003-1032. doi: 10.3934/dcds.2009.25.1003 [19] Hermen Jan Hupkes, Emmanuelle Augeraud-Véron. Well-posedness of initial value problems for functional differential and algebraic equations of mixed type. Discrete and Continuous Dynamical Systems, 2011, 30 (3) : 737-765. doi: 10.3934/dcds.2011.30.737 [20] Merab Svanadze. On the theory of viscoelasticity for materials with double porosity. Discrete and Continuous Dynamical Systems - B, 2014, 19 (7) : 2335-2352. doi: 10.3934/dcdsb.2014.19.2335

Impact Factor: