On Computational Aspects of a Regular n-Simplex Bisection

Branch-and-Bound (B&B) algorithms to solve Global Optimization (GO) may use n-simplicial partition sets. The n-simplex represents an n-dimensional body with n+1 vertices in (n+1)-dimensional space. The aim of this article is to investigate the properties of the binary tree generated by iterative bisection of the longest edge (LE) of the regular n-simplex as search space. This way of splitting an n-simplex reduces the appearance of bad shaped simplices which facilitates the convergence of the algorithm. It also helps to have a more uniform sampling of the search space since the function is evaluated at vertices of simplices. A motivation for this research is the estimation of the pending computational work load during the B&B GO algorithm. Such estimation may be helpful to steer parallel versions of the algorithms. In this paper we will show that the way the longest edge is selected affects on the number of sub-problems, the number of similar shapes of those sub-problems and their roundness factor. The computational cost to obtain those metrics increases with the dimension n. Here we show the results for n leq 3, where for n=3, we have a 3-dimensional body in a 4-dimensional space. Due to the exponential growth of the binary tree, high performance computing is useful in order to reach a high precision or when n eq 3. We make use of parallel computing under MATLAB software.