# American Institute of Mathematical Sciences

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

 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.

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 and 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. [2] J. I. Brauman, {Some histroical background on the double-well potential model}, Journal of Mass Spectrometry, 30 (1995), 1649-1651. [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. [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. [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] R. A. Horn and C. R. Johnson, Matrix Analysis Cambridge University Press, Cambridge, UK, 1985. doi: 10.1017/CBO9780511810817. [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. [8] J. J. Moré, Generalizations of the trust region problem, Optimization Methods & Software, 2 (1993), 189-209. [9] J. Nocedal and S. J. Wright, Numerical Optimization, 2nd edition, Springer, 2006. [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. [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.

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. [2] J. I. Brauman, {Some histroical background on the double-well potential model}, Journal of Mass Spectrometry, 30 (1995), 1649-1651. [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. [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. [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] R. A. Horn and C. R. Johnson, Matrix Analysis Cambridge University Press, Cambridge, UK, 1985. doi: 10.1017/CBO9780511810817. [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. [8] J. J. Moré, Generalizations of the trust region problem, Optimization Methods & Software, 2 (1993), 189-209. [9] J. Nocedal and S. J. Wright, Numerical Optimization, 2nd edition, Springer, 2006. [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. [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.
A double well potential problem having infinitely many local non-global minima
The graph of $g(w)$ in Example 1 ($n=1$)
The secular function (64)
The function $g(w)$ in Example 2 and its contour ($n=2$)
The secular function (65)
The function $g(w)$ in Example 3 and its contour ($n=2$)
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