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 & Management Optimization, 2017, 13 (3) : 1307-1328. doi: 10.3934/jimo.2016074
References:

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References:
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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