Double Well Potential Function and Its Optimization in the n-dimensional Real Space - Part II

TL;DR

This paper characterizes local and global extrema of double well potential functions in n-dimensional space, providing conditions and algorithms to identify critical points directly from the primal formulation.

Contribution

It introduces a direct primal approach to classify critical points of double well functions, including conditions for uniqueness and existence of local and global extrema.

Findings

At most one local non-global minimizer and one local maximizer exist for nonsingular functions.

Local maximizers are 'surrounded' by local minimizers in norm.

Three algorithms are proposed for identifying critical points.

Abstract

In contrast to taking the dual approach for finding a global minimum solution of a double well potential function, in Part II of the paper, we characterize a local minimizer, local maximizer, and global minimizer directly from the primal side. It is proven that, for a ``nonsingular" double well function, there exists at most one local, but non-global, minimizer and at most one local maximizer. Moreover, when it exists, the local maximizer is ``surrounded" by local minimizers in the sense that the norm of the local maximizer is strictly less than that of any local minimizer. We also establish some necessary and sufficient optimality conditions for the global minimizer, local non-global minimizer and local maximizer by studying a convex secular function over specific intervals. These conditions lead to three algorithms for identifying different types of critical points of a given double well function.