Cyclic Groups, Cutting Planes, Shortest Paths

Abstract

To define C(T), the convex closure of T, as the solution set of a system of linear inequalities, a class of functions II(f0) on the unit interval is defined. In this, f0 is any real number satisfying 0 < f0 < 1. This chapter presents a theorem that describes C(T) as an intersection of half-planes. In the algorithm, fixed breakpoints become convex and increasing breakpoints become concave breakpoints. The chapter discusses cyclic groups and cutting plane through alogorithms and presents some cyclic group and cutting plane problems and their solutions. The group problem, even when the group is not cyclic but more generally a direct product of cyclic groups, can be formulated as a shortest path problem. The chapter presents an algorithm that is a method of solving the shortest path problem and can be compared to other methods.