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

Shu-Cherng Fang; David Y. Gao; Gang-Xuan Lin; Ruey-Lin Sheu; Wenxun Xing

doi:10.3934/jimo.2016073

Article Contents

2017, Volume 13, Issue 3: 1291-1305. Doi: 10.3934/jimo.2016073

This issue Previous Article Optimal Sharpe ratio in continuous-time markets with and without a risk-free asset Next Article Double well potential function and its optimization in the

$N$

-dimensional real space-part Ⅱ

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: September 2017

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Abstract

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.

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

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Figure 1. Illustrative examples for the double well potential functions (DWP).

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Figure 2. The graph of $P(w)$ in Example 1 and the corresponding dual of the dual problem

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Figure 3. The graph of $P(w)$ in Example 2 and the corresponding dual of the dual problem

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Figure 4. The graph of $P(w)$ in Example 3 and the corresponding dual of the dual problem

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