The D₁-Triangulation in Variable Dimension Algorithms on the Unit Simplex

Abstract

Simplicial variable dimension algorithms to compute fixed points on the unit simplex were initiated by Kuhn in [98]. Kuhn’s variable dimension algorithm subdivides the unit simplex according to the Q-triangulation and starts at one of vertices of the unit simplex. It generates a sequence of adjacent simplices with varying dimension and terminates as soon as an approximate solution is yielded. If the accuracy is not good enough, one can restart Kuhn’s variable dimension algorithm at one of the vertices of the unit simplex with a finer simplicial subdivision. However, it is obvious that Kuhn’s algorithm loses all information about the location of a solution obtained in the previous implementation when restarting. To overcome this drawback, van der Laan and Talman originated a new generation of simplicial variable dimension algorithms on the unit simplex in [108]. Van der Laan and Talman’s variable dimension algorithm also subdivides the unit simplex according to the Q-triangulation, but it can start at an arbitrary grid point. Van der Laan and Talman’s algorithm leaves this grid point along one out of n + 1 different rays. It also generates a sequence of adjacent simplices with varying dimension and terminates in case that an approximate solution is obtained. If the accuracy is not high enough, one can restart the algorithm at the grid point closest to the approximate solution obtained in the previous implementation.