American Institute of Mathematical Sciences

Advanced
July  2017, 13(3): 1307-1328. doi: 10.3934/jimo.2016074

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

Yong Xia 1, , Ruey-Lin Sheu 2, , Shu-Cherng Fang 3, and  Wenxun Xing 4,

1. 

State Key Laboratory of Software Development Environment, School of Mathematics and System Sciences, Beihang University, China
2. 

Department of Mathematics, National Cheng Kung University, Taiwan
3. 

Department of Industrial and Systems Engineering, North Carolina State University, Raleigh, USA
4. 

Department of Mathematical Sciences, Tsinghua University, Beijing, China

Received  December 2015 Revised  August 2016 Published  October 2016

In contrast to taking the dual approach for finding a global minimum solution of a double well potential function, in Part Ⅱ of the paper, we characterize the local minimizer, local maximizer, and global minimizer directly from the primal side. It is proven that, for a ''nonsingular" double well function, there exists at most one local, but non-global, minimizer and at most one local maximizer. Moreover, the local maximizer is ''surrounded" by local minimizers in the sense that the norm of the local maximizer is strictly less than that of any local minimizer. We also establish necessary and sufficient optimality conditions for the global minimizer, local non-global minimizer and local maximizer by studying a convex secular function over specific intervals. These conditions lead to three algorithms for identifying different types of critical points of a given double well function.

Keywords: Double well potential, local minimizer, local maximizer, global minimum.
Mathematics Subject Classification: 49K30, 90C46, 90C26.
Citation: 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 & Management Optimization, 2017, 13 (3) : 1307-1328. doi: 10.3934/jimo.2016074
References:
[1]

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

[2]

J. I. Brauman, {Some histroical background on the double-well potential model}, Journal of Mass Spectrometry, 30 (1995), 1649-1651. Google Scholar

[3]

A. R. Conn, N. I. M. Gould and Ph. L. Toint, Trust-Region Methods Number 01, MPS-SIAM Series on Optimization, SIAM, Philadelphia, USA, 2000. doi: 10.1137/1.9780898719857.  Google Scholar

[4]

S. C. Fang, D. Y. Gao, G. X. Lin, R. L. Sheu and W. Xing, Double well potential function and its optimization in the n-dimensional real space -Part Ⅰ, Journal of Industrial and Management Optimization, in press, (2016).  doi: 10.3934/jimo.2016073.  Google Scholar

[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.  Google Scholar

[6]

R. A. Horn and C. R. Johnson, Matrix Analysis Cambridge University Press, Cambridge, UK, 1985. doi: 10.1017/CBO9780511810817.  Google Scholar

[7]

J. M. Martínez, Local minimizers of quadratic function on Euclidean balls and spheres, SIAM Journal on Optimization, 4 (1994), 159-176.  doi: 10.1137/0804009.  Google Scholar

[8]

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

[9]

J. Nocedal and S. J. Wright, Numerical Optimization, 2nd edition, Springer, 2006.  Google Scholar

[10]

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.  Google Scholar

[11]

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]

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

[2]

J. I. Brauman, {Some histroical background on the double-well potential model}, Journal of Mass Spectrometry, 30 (1995), 1649-1651. Google Scholar

[3]

A. R. Conn, N. I. M. Gould and Ph. L. Toint, Trust-Region Methods Number 01, MPS-SIAM Series on Optimization, SIAM, Philadelphia, USA, 2000. doi: 10.1137/1.9780898719857.  Google Scholar

[4]

S. C. Fang, D. Y. Gao, G. X. Lin, R. L. Sheu and W. Xing, Double well potential function and its optimization in the n-dimensional real space -Part Ⅰ, Journal of Industrial and Management Optimization, in press, (2016).  doi: 10.3934/jimo.2016073.  Google Scholar

[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.  Google Scholar

[6]

R. A. Horn and C. R. Johnson, Matrix Analysis Cambridge University Press, Cambridge, UK, 1985. doi: 10.1017/CBO9780511810817.  Google Scholar

[7]

J. M. Martínez, Local minimizers of quadratic function on Euclidean balls and spheres, SIAM Journal on Optimization, 4 (1994), 159-176.  doi: 10.1137/0804009.  Google Scholar

[8]

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

[9]

J. Nocedal and S. J. Wright, Numerical Optimization, 2nd edition, Springer, 2006.  Google Scholar

[10]

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.  Google Scholar

[11]

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.  A double well potential problem having infinitely many local non-global minima
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  The graph of $g(w)$ in Example 1 ($n=1$)
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  The secular function (64)
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.  The function $g(w)$ in Example 2 and its contour ($n=2$)
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 5.  The secular function (65)
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 6.  The function $g(w)$ in Example 3 and its contour ($n=2$)
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

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 & Continuous Dynamical Systems, 2019, 39 (5) : 2661-2678. doi: 10.3934/dcds.2019111
[2]

Sergey Zelik, Jon Pennant. Global well-posedness in uniformly local spaces for the Cahn-Hilliard equation in $\mathbb{R}^3$. Communications on Pure & Applied Analysis, 2013, 12 (1) : 461-480. doi: 10.3934/cpaa.2013.12.461
[3]

Keyan Wang. Global well-posedness for a transport equation with non-local velocity and critical diffusion. Communications on Pure & Applied Analysis, 2008, 7 (5) : 1203-1210. doi: 10.3934/cpaa.2008.7.1203
[4]

Francis Ribaud, Stéphane Vento. Local and global well-posedness results for the Benjamin-Ono-Zakharov-Kuznetsov equation. Discrete & Continuous Dynamical Systems, 2017, 37 (1) : 449-483. doi: 10.3934/dcds.2017019
[5]

Mohamad Darwich. Local and global well-posedness in the energy space for the dissipative Zakharov-Kuznetsov equation in 3D. Discrete & Continuous Dynamical Systems - B, 2020, 25 (9) : 3715-3724. doi: 10.3934/dcdsb.2020087
[6]

Rui L. Fernandes, Yuxuan Zhang. Local and global integrability of Lie brackets. Journal of Geometric Mechanics, 2021, 13 (3) : 355-384. doi: 10.3934/jgm.2021024
[7]

Boris Kolev. Local well-posedness of the EPDiff equation: A survey. Journal of Geometric Mechanics, 2017, 9 (2) : 167-189. doi: 10.3934/jgm.2017007
[8]

Wulong Liu, Guowei Dai. Multiple solutions for a fractional nonlinear Schrödinger equation with local potential. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2105-2123. doi: 10.3934/cpaa.2017104
[9]

J. W. Neuberger. How to distinguish a local semigroup from a global semigroup. Discrete & Continuous Dynamical Systems, 2013, 33 (11&12) : 5293-5303. doi: 10.3934/dcds.2013.33.5293
[10]

Stefano Galatolo. Global and local complexity in weakly chaotic dynamical systems. Discrete & Continuous Dynamical Systems, 2003, 9 (6) : 1607-1624. doi: 10.3934/dcds.2003.9.1607
[11]

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 & Optimization, 2013, 3 (2) : 271-282. doi: 10.3934/naco.2013.3.271
[12]

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
[13]

Yacheng Liu, Runzhang Xu. Potential well method for initial boundary value problem of the generalized double dispersion equations. Communications on Pure & Applied Analysis, 2008, 7 (1) : 63-81. doi: 10.3934/cpaa.2008.7.63
[14]

Christopher Henderson, Stanley Snelson, Andrei Tarfulea. Local well-posedness of the Boltzmann equation with polynomially decaying initial data. Kinetic & Related Models, 2020, 13 (4) : 837-867. doi: 10.3934/krm.2020029
[15]

Yong Zhou, Jishan Fan. Local well-posedness for the ideal incompressible density dependent magnetohydrodynamic equations. Communications on Pure & Applied Analysis, 2010, 9 (3) : 813-818. doi: 10.3934/cpaa.2010.9.813
[16]

Caochuan Ma, Zaihong Jiang, Renhui Wan. Local well-posedness for the tropical climate model with fractional velocity diffusion. Kinetic & Related Models, 2016, 9 (3) : 551-570. doi: 10.3934/krm.2016006
[17]

Timur Akhunov. Local well-posedness of quasi-linear systems generalizing KdV. Communications on Pure & Applied Analysis, 2013, 12 (2) : 899-921. doi: 10.3934/cpaa.2013.12.899
[18]

Hung Luong. Local well-posedness for the Zakharov system on the background of a line soliton. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2657-2682. doi: 10.3934/cpaa.2018126
[19]

Hartmut Pecher. Local well-posedness for the nonlinear Dirac equation in two space dimensions. Communications on Pure & Applied Analysis, 2014, 13 (2) : 673-685. doi: 10.3934/cpaa.2014.13.673
[20]

Jae Min Lee, Stephen C. Preston. Local well-posedness of the Camassa-Holm equation on the real line. Discrete & Continuous Dynamical Systems, 2017, 37 (6) : 3285-3299. doi: 10.3934/dcds.2017139

2020 Impact Factor: 1.801

Tools

Metrics

  • PDF downloads (82)
  • HTML views (384)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy