Three-point bounds for energy minimization
Three-point bounds for energy minimization
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DOI：
10.1090/s0894-0347-2012-00737-1
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发表时间：
2011-03
影响因子：
3.9
通讯作者：
Henry Cohn;Jeechul Woo
中科院分区：
文献类型：
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作者：
Henry Cohn;Jeechul Woo
三点半定规划界是已知的最强大的球码大小的界工具之一。在这篇文章中，我们用它们来证明粒子通过一对势函数相互作用的势能的下界。证明了我们的界对于RP^2中的7个点是尖锐的。具体地说，我们证明了连接立方体及其对偶八面体的相对顶点的7条直线是全局最优的。(换句话说，在通过原点的七条直线的所有构型中，这一种构型使所有势函数的能量最小化，这些势函数是弦距离平方的完全单调函数。)这种构型是已知的唯一不是距离正则的泛最优解，也是RP^2中最后一个剩余的泛最优解。我们还给出了半定规划界的一个新的推导和关于它们的几个令人惊讶的猜想。
Three-point semidefinite programming bounds are one of the most powerful known tools for bounding the size of spherical codes. In this paper, we use them to prove lower bounds for the potential energy of particles interacting via a pair potential function. We show that our bounds are sharp for seven points in RP^2. Specifically, we prove that the seven lines connecting opposite vertices of a cube and of its dual octahedron are universally optimal. (In other words, among all configurations of seven lines through the origin, this one minimizes energy for all potential functions that are completely monotonic functions of squared chordal distance.) This configuration is the only known universal optimum that is not distance regular, and the last remaining universal optimum in RP^2. We also give a new derivation of semidefinite programming bounds and present several surprising conjectures about them.