# American Institute of Mathematical Sciences

July  2017, 13(3): 1291-1305. doi: 10.3934/jimo.2016073

## Double well potential function and its optimization in the $N$ -dimensional real space-part Ⅰ

 1 Department of Industrial and Systems Engineering, North Carolina State University, Raleigh, USA 2 School of Science, Information Technology, and Engineering, Federation University Australia, Mt Helen, Australia 3 Department of Mathematics, National Cheng Kung University, Taiwan 4 Department of Mathematical Sciences, Tsinghua University, Beijing, China

Received  December 2015 Revised  August 2016 Published  October 2016

A special type of multi-variate polynomial of degree 4, called the double well potential function, is studied. It is derived from a discrete approximation of the generalized Ginzburg-Landau functional, and we are interested in understanding its global minimum solution and all local non-global points. The main difficulty for the model is due to its non-convexity. In part Ⅰ of the paper, we first characterize the global minimum solution set, whereas the study for local non-global optimal solutions is left for Part Ⅱ. We show that, the dual of the Lagrange dual of the double well potential problem is a linearly constrained convex minimization problem, which, under a designated nonlinear transformation, can be equivalently mapped to a portion of the original double well potential function containing the global minimum. In other words, solving the global minimum of the double well potential function is essentially a convex minimization problem, despite of its non-convex nature. Numerical examples are provided to illustrate the important features of the problem and the mapping in between.

Citation: 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
##### References:
 [1] M. S. Bazaraa, H. D. Sherali and C. M. Shetty, Nonlinear Programming: Theory and Algorithms 3rd. , Wiley Interscience, New York, 2006. doi: 10.1002/0471787779. [2] A. Ben-Tal and M. Teboulle, Hidden convexity in some nonconvex quadratically constrained quadratic programming, Mathematical Programming, 72 (1996), 51-63.  doi: 10.1007/BF02592331. [3] T. Bidoneau, On the Van Der Waals theory of surface tension, Markov Processes and Related Fields, 8 (2002), 319-338. [4] J. I. Brauman, Some historical background on the double-well potential model, Journal of Mass Spectrometry, 30 (1995), 1649-1651.  doi: 10.1002/jms.1190301203. [5] J. M. Feng, G. X. Lin, R. L. Sheu and Y. Xia, Duality and solutions for quadratic programming over single non-homogeneous quadratic constraint, Journal of Global Optimization, 54 (2012), 275-293.  doi: 10.1007/s10898-010-9625-6. [6] D. Y. Gao and G. Strang, Geometrical nonlinearity: Potential energy, complementary energy, and the gap function, Quarterly of Applied Mathematics, 47 (1989), 487-504. [7] D. Y. Gao, Duality Principles in Nonconvex Systems: Theory, Methods and Applications Kluwer Academic, Dordrecht, 2000. doi: 10.1007/978-1-4757-3176-7. [8] D. Y. Gao and H. Yu, Multi-scale modelling and canonical dual finite element method in phase transitions of solids, International Journal of Solids and Structures, 45 (2008), 3660-3673.  doi: 10.1016/j.ijsolstr.2007.08.027. [9] A. Heuer nad U. Haeberlen, The dynamics of hydrogens in double well potentials: The transition of the jump rate from the low temperature quantum-mechanical to the high temperature activated regime, Journal of Chemical Physics, 95 (1991), 4201-4214. [10] H. C. Hu, On some variational principles in the theory of elasticity and the theory of plasticity, Scientia Sinica, 4 (1995), 33-54. [11] R. L. Jerrard, Lower bounds for generalized Ginzburg-Landau functionals, SIAM Journal on Mathematical Analysis, 30 (1999), 721-746.  doi: 10.1137/S0036141097300581. [12] K. Kaski, K. Binder and J. D. Gunton, A study of a coarse-gained free energy funcitonal for the three-dimensional Ising model, Journal of Physics A: Mathematical and General, 16 (1983), 623-627. [13] J. J. Moré, Generalizations of the trust region problem, Optimization Methods & Software, 2 (1993), 189-209. [14] K. Washizu, On the variational principle for elascticity and plasticity, Technical Report, Aeroelastic and Structures Research Laboratery, MIT, Cambridge, (1966), 25-18. [15] Y. Xia, S. Wang and R. L. Sheu, S-lemma with equality and its applications, Mathematical Programming, 156 (2016), 513-547.  doi: 10.1007/s10107-015-0907-0. [16] W. Xing, S. C. Fang, D. Y. Gao, R. L. Sheu and L. Zhang, Canonical dual solutions to the quadratic programming problem over a quadratic constraint, Asia-Pacific Journal of Operational Research, 32 (2015), 1540007.

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##### References:
 [1] M. S. Bazaraa, H. D. Sherali and C. M. Shetty, Nonlinear Programming: Theory and Algorithms 3rd. , Wiley Interscience, New York, 2006. doi: 10.1002/0471787779. [2] A. Ben-Tal and M. Teboulle, Hidden convexity in some nonconvex quadratically constrained quadratic programming, Mathematical Programming, 72 (1996), 51-63.  doi: 10.1007/BF02592331. [3] T. Bidoneau, On the Van Der Waals theory of surface tension, Markov Processes and Related Fields, 8 (2002), 319-338. [4] J. I. Brauman, Some historical background on the double-well potential model, Journal of Mass Spectrometry, 30 (1995), 1649-1651.  doi: 10.1002/jms.1190301203. [5] J. M. Feng, G. X. Lin, R. L. Sheu and Y. Xia, Duality and solutions for quadratic programming over single non-homogeneous quadratic constraint, Journal of Global Optimization, 54 (2012), 275-293.  doi: 10.1007/s10898-010-9625-6. [6] D. Y. Gao and G. Strang, Geometrical nonlinearity: Potential energy, complementary energy, and the gap function, Quarterly of Applied Mathematics, 47 (1989), 487-504. [7] D. Y. Gao, Duality Principles in Nonconvex Systems: Theory, Methods and Applications Kluwer Academic, Dordrecht, 2000. doi: 10.1007/978-1-4757-3176-7. [8] D. Y. Gao and H. Yu, Multi-scale modelling and canonical dual finite element method in phase transitions of solids, International Journal of Solids and Structures, 45 (2008), 3660-3673.  doi: 10.1016/j.ijsolstr.2007.08.027. [9] A. Heuer nad U. Haeberlen, The dynamics of hydrogens in double well potentials: The transition of the jump rate from the low temperature quantum-mechanical to the high temperature activated regime, Journal of Chemical Physics, 95 (1991), 4201-4214. [10] H. C. Hu, On some variational principles in the theory of elasticity and the theory of plasticity, Scientia Sinica, 4 (1995), 33-54. [11] R. L. Jerrard, Lower bounds for generalized Ginzburg-Landau functionals, SIAM Journal on Mathematical Analysis, 30 (1999), 721-746.  doi: 10.1137/S0036141097300581. [12] K. Kaski, K. Binder and J. D. Gunton, A study of a coarse-gained free energy funcitonal for the three-dimensional Ising model, Journal of Physics A: Mathematical and General, 16 (1983), 623-627. [13] J. J. Moré, Generalizations of the trust region problem, Optimization Methods & Software, 2 (1993), 189-209. [14] K. Washizu, On the variational principle for elascticity and plasticity, Technical Report, Aeroelastic and Structures Research Laboratery, MIT, Cambridge, (1966), 25-18. [15] Y. Xia, S. Wang and R. L. Sheu, S-lemma with equality and its applications, Mathematical Programming, 156 (2016), 513-547.  doi: 10.1007/s10107-015-0907-0. [16] W. Xing, S. C. Fang, D. Y. Gao, R. L. Sheu and L. Zhang, Canonical dual solutions to the quadratic programming problem over a quadratic constraint, Asia-Pacific Journal of Operational Research, 32 (2015), 1540007.
Illustrative examples for the double well potential functions (DWP).
The graph of $P(w)$ in Example 1 and the corresponding dual of the dual problem
The graph of $P(w)$ in Example 2 and the corresponding dual of the dual problem
The graph of $P(w)$ in Example 3 and the corresponding dual of the dual problem
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