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

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
New sufficient global optimality conditions for linearly constrained bivalent quadratic optimization problems
Yong Xia
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
Convex hull of the orthogonal similarity set with applications in quadratic assignment problems
Yong Xia
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
Some new classes of cyclic codes with three or six weights
Yongbo Xia Tor Helleseth Chunlei Li
In this paper, a class of three-weight cyclic codes over prime fields $\mathbb{F}_p$ of odd order whose duals have two zeros, and a class of six-weight cyclic codes whose duals have three zeros are presented. The weight distributions of these cyclic codes are derived.
keywords: exponential sum weight distribution Cyclic code quadratic form.
A new semidefinite relaxation for $L_{1}$-constrained quadratic optimization and extensions
Yong Xia Yu-Jun Gong Sheng-Nan Han
In this paper, by improving the variable-splitting approach, we propose a new semidefinite programming (SDP) relaxation for the nonconvex quadratic optimization problem over the $\ell_1$ unit ball (QPL1). It dominates the state-of-the-art SDP-based bound for (QPL1). As extensions, we apply the new approach to the relaxation problem of the sparse principal component analysis and the nonconvex quadratic optimization problem over the $\ell_p$ ($1< p<2$) unit ball and then show the dominance of the new relaxation.
keywords: $\ell_1$ unit ball Semidefinite programming Sparse principal component analysis. Quadratic optimization

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