A Simplicial Algorithm for Computing Robust Stationary Points of a Continuous Function on the Unit Simplex

Abstract

A simplicial algorithm is proposed to compute a robust stationary point of a continuous function f from the $(n - 1)$-dimensional unit simplex $S^{n - 1} $ into $R^n $. The concept of robust stationary point is a refinement of the concept of stationary point of f on $S^{n - 1} $. Starting from an arbitrarily chosen interior point v in $S^{n - 1} $, the algorithm generates a piecewise linear path of points in $S^{n - 1} $. This path is followed by alternating linear programming pivot steps and replacement steps in a specific simplicial subdivision of the relative interior of $S^{n - 1} $. In this way an approximate robust stationary point of any a priori chosen accuracy is reached within a finite number of steps. The algorithm leaves the starting point along one out of $n!$ rays. When the path approaches the boundary of $S^{n - 1} $, the mesh size of the triangulation along the path automatically goes to zero. This makes the algorithm different from all simplicial restart algorithms and homotopy algorithms known so far. Roughly speaking, the algorithm is a blend of a restart and a homotopy algorithm and maintains the basic properties of both. However, the algorithm does not need an extra dimension as homotopy algorithms do. Some examples are discussed.