# American Institute of Mathematical Sciences

July  2017, 13(3): 1291-1305. doi: 10.3934/jimo.2016073

## 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  October 2016

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.

Citation: 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
##### References:

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##### References:
Illustrative examples for the double well potential functions (DWP).
The graph of $P(w)$ in Example 1 and the corresponding dual of the dual problem
The graph of $P(w)$ in Example 2 and the corresponding dual of the dual problem
The graph of $P(w)$ in Example 3 and the corresponding dual of the dual problem
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