Double well potential function and its optimization in the \begin{document}$N$\end{document} -dimensional real space-part Ⅰ
| Autor: | Wenxun Xing, Ruey-Lin Sheu, Shu-Cherng Fang, David Yang Gao, Gang-Xuan Lin |
|---|---|
| Rok vydání: | 2017 |
| Předmět: |
Polynomial
021103 operations research Control and Optimization Degree (graph theory) Applied Mathematics Strategy and Management 0211 other engineering and technologies Solution set 02 engineering and technology Function (mathematics) Type (model theory) Space (mathematics) Atomic and Molecular Physics and Optics Dual (category theory) Convex optimization 0202 electrical engineering electronic engineering information engineering Applied mathematics 020201 artificial intelligence & image processing Business and International Management Electrical and Electronic Engineering Mathematics |
| Zdroj: | Journal of Industrial & Management Optimization. 13:1291-1305 |
| ISSN: | 1553-166X |
| DOI: | 10.3934/jimo.2016073 |
| Popis: | 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. |
| Databáze: | OpenAIRE |
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