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

• 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.

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

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

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

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

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

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