Adjacency on Convex Polyhedra

Abstract

Adjacency properties of extreme points of a convex polyhedron are discussed. In mathematical programming we are quite often faced with problems of characterizing the extreme points of a convex polyhedron of the form $K \cap H$ or $K \cap L$, where $K \subset R^n $ is a convex polyhedron whose extreme points and their adjacency structure is known, $H \subset R^n $ is a hyperplane and $L \subset R^n $ is a closed half-space. This has been done in § 4. Connectedness properties of a pair of extreme points of K in L by edge paths of K lying in L are discussed in § 6.Suppose $K \subset R^n $ is a convex polytope. Then K is the convex hull of its extreme points. A collapsed polytope ofK is the convex hull of a proper subset of the set of extreme points of K. Several hard problems in mathematical programming, like the traveling salesman problem, the 0–1 mixed integer programming problem etc. deal with optimization on collapsed polytopes of polytopes determined by few simple constraints. Adjacency properties of extreme points on collapsed polytopes are discussed in terms of adjacency and edge paths on the original polytope in § 5.