Generating a smallest binary tree by proper selection of the longest edges to bisect in a unit simplex refinement

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• 作者：J. M. G. Salmerón ; G. Aparicio ; L. G. Casado…
• 关键词：Regular simplex ; Longest edge bisection ; Branch ; and ; bound ; Bisection sequence ; Combinatorial optimization
• 刊名：Journal of Combinatorial Optimization
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：33
• 期：2
• 页码：389-402
• 全文大小：661KB
• 刊物类别：Mathematics and Statistics
• 刊物主题：Combinatorics; Convex and Discrete Geometry; Mathematical Modeling and Industrial Mathematics; Theory of Computation; Optimization; Operation Research/Decision Theory;
• 出版者：Springer US
• ISSN：1573-2886
• 卷排序：33

文摘

In several areas like global optimization using branch-and-bound methods for mixture design, the unit n-simplex is refined by longest edge bisection (LEB). This process provides a binary search tree. For \(n>2\), simplices appearing during the refinement process can have more than one longest edge (LE). The size of the resulting binary tree depends on the specific sequence of bisected longest edges. The questions are how to calculate the size of one of the smallest binary trees generated by LEB and how to find the corresponding sequence of LEs to bisect, which can be represented by a set of LE indices. Algorithms answering these questions are presented here. We focus on sets of LE indices that are repeated at a level of the binary tree. A set of LEs was presented in Aparicio et al. (Informatica 26(1):17–32, 2015), for \(n=3\). An additional question is whether this set is the best one under the so-called \(m_k\)-valid condition.