JIMO
Double well potential function and its optimization in the $N$ -dimensional real space-part Ⅱ
Yong Xia Ruey-Lin Sheu Shu-Cherng Fang Wenxun Xing
Journal of Industrial & Management Optimization 2017, 13(3): 1307-1328 doi: 10.3934/jimo.2016074

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 the 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, 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 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.

keywords: Double well potential local minimizer local maximizer global minimum
JIMO
New sufficient global optimality conditions for linearly constrained bivalent quadratic optimization problems
Yong Xia
Journal of Industrial & Management Optimization 2009, 5(4): 881-892 doi: 10.3934/jimo.2009.5.881
In this article, we obtain new sufficient global optimality conditions for bivalent quadratic optimization problems with linearly (equivalent and inequivalent) constraints, by exploring the local optimality condition. The global optimality condition can be further simplified when applied to special cases such as the $p$-dispersion-sum problem and the quadratic assignment problem.
keywords: optimality condition 0-1 quadratic programming quadratic assignment problem. integer programming
JIMO
Convex hull of the orthogonal similarity set with applications in quadratic assignment problems
Yong Xia
Journal of Industrial & Management Optimization 2013, 9(3): 689-701 doi: 10.3934/jimo.2013.9.689
In this paper, we study thoroughly the convex hull of the orthogonal similarity set and give a new representation. When applied in quadratic assignment problems, it motivates two new lower bounds. The first is equivalent to the projected eigenvalue bound, while the second highly outperforms several well-known lower bounds in literature.
keywords: semidefinite programming. Orthogonal similarity quadratic assignment problem convex hull lower bound

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