- Title
- Double well potential function and its optimization in the N-dimensional real space -- Part I
- Creator
- Fang, Shucherng; Gao, David; Lin, Gang-Xuan; Sheu, Ruey-Lin; Xing, Wenxun
- Date
- 2017
- Type
- Text; Journal article
- Identifier
- http://researchonline.federation.edu.au/vital/access/HandleResolver/1959.17/158516
- Identifier
- vital:11792
- Identifier
-
https://doi.org/10.3934/jimo.2016073
- Identifier
- ISSN:1547-5816
- 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.
- Relation
- Journal of Industrial and Management Optimization Vol. 13, no. 3 (2017), p. 1291-1305
- Rights
- © 2017 American Institute of Mathematical Sciences
- Rights
- This metadata is freely available under a CCO license
- Subject
- 0102 Applied Mathematics; 0103 Numerical and Computational Mathematics; Non-convex quadratic programming; Polynomial optimization; Generalized Ginzburg-Landau functional; Double well potential
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