Enhancements and computational evaluation of the hit-and-run random walk on polyhedra

Abstract

The symmetry function of a convex set offers us numerous useful information about the set in relation to probabilistic theory and geometric properties. The symmetry function is a measure of how symmetric the convex set is, and for a point, intuitively it measures how symmetric the set is with respect to that point. We call a point of high symmetry value a deep point. A random walk is a procedure that starts from a particular point in Rn and at each iteration, moves to a "neighboring" point according to some probability distribution that depends solely on the current point. The Hit-and-Run random walk on a convex set S picks a random line e through the point, and at next iteration goes to a new point that is chosen uniformly on the chord ℓ [intersection] S. In this thesis, we analyze and investigate the effectiveness of the Hit-and-Run random walk to compute a deep point in a convex body, given a randomly generated convex set. The effectiveness is evaluated in terms of the role of the starting point and the likelihood that the random walk will enter the zone of high symmetry. Additionally, some known probabilistic properties of the symmetry function are tested using the random walk, from which the integrity of the code is also verified.

(cont.) The final portion of this thesis analyzes the behavioral properties of convex sets that have non-Euclidean rounding, which renders the random walk less efficient. Therefore the pre-conditioned Hit-and-Run random walk is performed, and the performance is quantitatively presented in a power law equation that predicts the preconditioning iterations required, given the dimension of the convex set, a starting point near a corner and the width of that corner.