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. 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.

[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. 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. 

[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. 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.

[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.

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.

[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. 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.

[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. 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. 

[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. 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.

[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.

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
[1]

Yong Wang, Wanquan Liu, Guanglu Zhou. An efficient algorithm for non-convex sparse optimization. Journal of Industrial and Management Optimization, 2019, 15 (4) : 2009-2021. doi: 10.3934/jimo.2018134

[2]

Noboru Okazawa, Tomomi Yokota. Smoothing effect for generalized complex Ginzburg-Landau equations in unbounded domains. Conference Publications, 2001, 2001 (Special) : 280-288. doi: 10.3934/proc.2001.2001.280

[3]

Simão Correia, Mário Figueira. A generalized complex Ginzburg-Landau equation: Global existence and stability results. Communications on Pure and Applied Analysis, 2021, 20 (5) : 2021-2038. doi: 10.3934/cpaa.2021056

[4]

Nurullah Yilmaz, Ahmet Sahiner. On a new smoothing technique for non-smooth, non-convex optimization. Numerical Algebra, Control and Optimization, 2020, 10 (3) : 317-330. doi: 10.3934/naco.2020004

[5]

Hans G. Kaper, Bixiang Wang, Shouhong Wang. Determining nodes for the Ginzburg-Landau equations of superconductivity. Discrete and Continuous Dynamical Systems, 1998, 4 (2) : 205-224. doi: 10.3934/dcds.1998.4.205

[6]

Mickaël Dos Santos, Oleksandr Misiats. Ginzburg-Landau model with small pinning domains. Networks and Heterogeneous Media, 2011, 6 (4) : 715-753. doi: 10.3934/nhm.2011.6.715

[7]

Fanghua Lin, Ping Zhang. On the hydrodynamic limit of Ginzburg-Landau vortices. Discrete and Continuous Dynamical Systems, 2000, 6 (1) : 121-142. doi: 10.3934/dcds.2000.6.121

[8]

Dingshi Li, Xiaohu Wang. Asymptotic behavior of stochastic complex Ginzburg-Landau equations with deterministic non-autonomous forcing on thin domains. Discrete and Continuous Dynamical Systems - B, 2019, 24 (2) : 449-465. doi: 10.3934/dcdsb.2018181

[9]

Hong Lu, Mingji Zhang. Dynamics of non-autonomous fractional Ginzburg-Landau equations driven by colored noise. Discrete and Continuous Dynamical Systems - B, 2020, 25 (9) : 3553-3576. doi: 10.3934/dcdsb.2020072

[10]

Lingyu Li, Zhang Chen. Asymptotic behavior of non-autonomous random Ginzburg-Landau equation driven by colored noise. Discrete and Continuous Dynamical Systems - B, 2021, 26 (6) : 3303-3333. doi: 10.3934/dcdsb.2020233

[11]

Yun Lan, Ji Shu. Dynamics of non-autonomous fractional stochastic Ginzburg-Landau equations with multiplicative noise. Communications on Pure and Applied Analysis, 2019, 18 (5) : 2409-2431. doi: 10.3934/cpaa.2019109

[12]

Feng Zhou, Chunyou Sun. Dynamics for the complex Ginzburg-Landau equation on non-cylindrical domains I: The diffeomorphism case. Discrete and Continuous Dynamical Systems - B, 2016, 21 (10) : 3767-3792. doi: 10.3934/dcdsb.2016120

[13]

Bo You, Yanren Hou, Fang Li, Jinping Jiang. Pullback attractors for the non-autonomous quasi-linear complex Ginzburg-Landau equation with $p$-Laplacian. Discrete and Continuous Dynamical Systems - B, 2014, 19 (6) : 1801-1814. doi: 10.3934/dcdsb.2014.19.1801

[14]

Lipeng Duan, Jun Yang. On the non-degeneracy of radial vortex solutions for a coupled Ginzburg-Landau system. Discrete and Continuous Dynamical Systems, 2021, 41 (10) : 4767-4790. doi: 10.3934/dcds.2021056

[15]

Dan Zhu, Rosemary A. Renaut, Hongwei Li, Tianyou Liu. Fast non-convex low-rank matrix decomposition for separation of potential field data using minimal memory. Inverse Problems and Imaging, 2021, 15 (1) : 159-183. doi: 10.3934/ipi.2020076

[16]

Luigi Forcella, Kazumasa Fujiwara, Vladimir Georgiev, Tohru Ozawa. Local well-posedness and blow-up for the half Ginzburg-Landau-Kuramoto equation with rough coefficients and potential. Discrete and Continuous Dynamical Systems, 2019, 39 (5) : 2661-2678. doi: 10.3934/dcds.2019111

[17]

Qilin Wang, Liu He, Shengjie Li. Higher-order weak radial epiderivatives and non-convex set-valued optimization problems. Journal of Industrial and Management Optimization, 2019, 15 (2) : 465-480. doi: 10.3934/jimo.2018051

[18]

Meixia Li, Changyu Wang, Biao Qu. Non-convex semi-infinite min-max optimization with noncompact sets. Journal of Industrial and Management Optimization, 2017, 13 (4) : 1859-1881. doi: 10.3934/jimo.2017022

[19]

Tong Li, Hui Yin. Convergence rate to strong boundary layer solutions for generalized BBM-Burgers equations with non-convex flux. Communications on Pure and Applied Analysis, 2014, 13 (2) : 835-858. doi: 10.3934/cpaa.2014.13.835

[20]

Hong Lu, Ji Li, Mingji Zhang. Stochastic dynamics of non-autonomous fractional Ginzburg-Landau equations on $ \mathbb{R}^3 $. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022028

2021 Impact Factor: 1.411

Metrics

  • PDF downloads (119)
  • HTML views (411)
  • Cited by (4)

[Back to Top]