On two problems over polyhedral convex sets

In this paper, we are concerned with the following two problems often encountered in concave programming: Given the vertices and extreme detections of a polyhedral convex set Modefined by a system of linear constraints, determine the vertices and extreme directions of the polyhedral convex set obtained from M just by adding one new linear equality (or inequality) constraint.Among the constraints of a given polyhedral convex set, find those which are redundant, i.e. which can be removed without affecting the polyhedral convex set.

On the Extreme Rays of the Metric Cone

A classical result in the theory of convex polyhedra is that every bounded polyhedral convex set can be expressed either as the intersection of half-spaces or as a convex combination of extreme points. It is becoming increasingly apparent that a full understanding of a class of convex polyhedra requires the knowledge of both of these characterizations. Perhaps the earliest and neatest example of this is the class of doubly stochastic matrices. This polyhedron can be defined by the system of equationsBirkhoff [2] and Von Neuman have shown that the extreme points of this bounded polyhedron are just the n × n permutation matrices. The importance of this result for mathematical programming is that it tells us that the maximum of any linear form over P will occur for a permutation matrix X.