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

Shu-Cherng Fang; David Y. Gao; Gang-Xuan Lin; Ruey-Lin Sheu; Wenxun Xing

doi:10.3934/jimo.2016073

Article Contents

2017, Volume 13, Issue 3: 1291-1305. Doi: 10.3934/jimo.2016073

This issue Previous Article Optimal Sharpe ratio in continuous-time markets with and without a risk-free asset Next Article Double well potential function and its optimization in the

$N$

-dimensional real space-part Ⅱ

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 online: October 2016

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Abstract

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.

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Mathematics Subject Classification: 49K30, 90C46, 90C26.

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Figure 1. Illustrative examples for the double well potential functions (DWP).

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Figure 2. The graph of $P(w)$ in Example 1 and the corresponding dual of the dual problem

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Figure 3. The graph of $P(w)$ in Example 2 and the corresponding dual of the dual problem

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Figure 4. The graph of $P(w)$ in Example 3 and the corresponding dual of the dual problem

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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.