Algorithms for on-line vertex enumeration problem

Abstract

Vertex enumeration problem is to enumerate all vertices of a polyhedron P which is given by intersection of finitely many halfspaces. It is a basis for many algorithms designed to solve problems from various application areas and there are many algorithms to solve these problems in the literature. On the one hand, there are iterative algorithms which solve the so called 'on-line' vertex enumeration problem in each iteration. In other words, in each iteration of these algorithms, the current polyhedron is intersected with an additional halfspace that defines P. On the other hand, there are simplex-type algorithms which takes the set off all halfspaces as its input from the beginning. One of the usages of the vertex enumeration problem is the Benson-type multiobjective optimization algorithms. The aim of these algorithms is to generate or approximate the Pareto frontier (the set of nondominated points in the objective space). In each iteration of the Benson's algorithm, a polyhedron which contains the Pareto frontier is intersected with an additional halfspace in order tofind a finer outer approximation. The vertex enumeration problem to be used within this algorithm has a special structure. Namely, the polyhedron to be generated is known to be unbounded with a recession cone which is equal to the positive orthant. In this thesis, we consider the double description method which is a method to solve an on-line vertex enumeration problem where the starting polyhedron is bounded. (1) We generate an iterative algorithm to solve the vertex enumeration problem from the scratch where polyhedron P is allowed to be bounded or unbounded. (2) Then, we slightly modify this algorithm to be more efficient while it only works for problems where the recession cone of P is known to be the positive orthant. (3) Finally, we generate an additional algorithm for these problems. For this one, we modify the double description method such that it uses the extreme directions of the recession cone more effectively. We provide an illustrative example to explain the algorithms in detail. We implement the algorithms using MATLAB; employ each of them as a function of a Benson-type multiobjective optimization algorithm; and test the performances of the algorithms for randomly generated linear multiobjective optimization problems. Accordingly, for two dimensional problems, there is no clear distinction between the run-time performances of these algorithms. However, as the dimension of the vertex enumeration problem increases, the last algorithm (Algorithm 3) gets more efficient than the others.