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

Yong Xia; Ruey-Lin Sheu; Shu-Cherng Fang; Wenxun Xing

doi:10.3934/jimo.2016074

Article Contents

2017, Volume 13, Issue 3: 1307-1328. Doi: 10.3934/jimo.2016074

This issue Previous Article Double well potential function and its optimization in the

$N$

-dimensional real space-part Ⅰ Next Article Single server retrial queues with setup time

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: November 30, 2015

Revised: July 31, 2016

Published: September 2017

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Abstract

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.

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Mathematics Subject Classification: 49K30, 90C46, 90C26.

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Figure 1. A double well potential problem having infinitely many local non-global minima

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

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Figure 3. The secular function (64)

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

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Figure 5. The secular function (65)

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

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