Pub. online:1 Jan 2016Type:Research ArticleOpen Access
Volume 27, Issue 2 (2016), pp. 351–366
The iterative bisection of the longest edge of the unit simplex generates a binary tree, where the specific shape depends on the chosen longest edges to be bisected. In global optimization, the use of various distance norms may be advantageous for bounding purposes. The question dealt with in this paper is how the size of a binary tree generated by the refinement process depends on heuristics for longest edge selection when various distance norms are used. We focus on the minimum size of the tree that can be reached, how selection criteria may reduce the size of the tree compared to selecting the first edge, whether a predefined grid is covered and how unique are the selection criteria. The exact numerical values are provided for the unit simplex in 4 and 5-dimensional space.
Volume 26, Issue 1 (2015), pp. 17–32
In several areas like Global Optimization using branch-and-bound methods, the unit n-simplex is refined by bisecting the longest edge such that a binary search tree appears. This process generates simplices belonging to different shape classes. Having less simplex shapes facilitates the prediction of the further workload from a node in the binary tree, because the same shape leads to the same sub-tree. Irregular sub-simplices generated in the refinement process may have more than one longest edge when . The question is how to choose the longest edge to be bisected such that the number of shape classes is as small as possible. We develop a Branch-and-Bound (B&B) algorithm to find the minimum number of classes in the refinement process. The developed B&B algorithm provides a minimum number of eight classes for a regular 3-simplex. Due to the high computational cost of solving this combinatorial problem, future research focuses on using high performance computing to derive the minimum number of shapes in higher dimensions.