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 & 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.  Google Scholar

[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.  Google Scholar

[3]

T. Bidoneau, On the Van Der Waals theory of surface tension, Markov Processes and Related Fields, 8 (2002), 319-338.   Google Scholar

[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.  Google Scholar

[5]

J. M. FengG. X. LinR. 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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[10]

H. C. Hu, On some variational principles in the theory of elasticity and the theory of plasticity, Scientia Sinica, 4 (1995), 33-54.   Google Scholar

[11]

R. L. Jerrard, Lower bounds for generalized Ginzburg-Landau functionals, SIAM Journal on Mathematical Analysis, 30 (1999), 721-746.  doi: 10.1137/S0036141097300581.  Google Scholar

[12]

K. KaskiK. 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.   Google Scholar

[13]

J. J. Moré, Generalizations of the trust region problem, Optimization Methods & Software, 2 (1993), 189-209.   Google Scholar

[14]

K. Washizu, On the variational principle for elascticity and plasticity, Technical Report, Aeroelastic and Structures Research Laboratery, MIT, Cambridge, (1966), 25-18.   Google Scholar

[15]

Y. XiaS. 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.  Google Scholar

[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. Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[3]

T. Bidoneau, On the Van Der Waals theory of surface tension, Markov Processes and Related Fields, 8 (2002), 319-338.   Google Scholar

[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.  Google Scholar

[5]

J. M. FengG. X. LinR. 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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[10]

H. C. Hu, On some variational principles in the theory of elasticity and the theory of plasticity, Scientia Sinica, 4 (1995), 33-54.   Google Scholar

[11]

R. L. Jerrard, Lower bounds for generalized Ginzburg-Landau functionals, SIAM Journal on Mathematical Analysis, 30 (1999), 721-746.  doi: 10.1137/S0036141097300581.  Google Scholar

[12]

K. KaskiK. 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.   Google Scholar

[13]

J. J. Moré, Generalizations of the trust region problem, Optimization Methods & Software, 2 (1993), 189-209.   Google Scholar

[14]

K. Washizu, On the variational principle for elascticity and plasticity, Technical Report, Aeroelastic and Structures Research Laboratery, MIT, Cambridge, (1966), 25-18.   Google Scholar

[15]

Y. XiaS. 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.  Google Scholar

[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. Google Scholar

Figure 1.  Illustrative examples for the double well potential functions (DWP).
Figure 2.  The graph of $P(w)$ in Example 1 and the corresponding dual of the dual problem
Figure 3.  The graph of $P(w)$ in Example 2 and the corresponding dual of the dual problem
Figure 4.  The graph of $P(w)$ in Example 3 and the corresponding dual of the dual problem
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