A Method to Solve the Neighbourhood Problem

Abstract

This chapter deals with a special degeneracy problem, namely the problem of determining all neighbouring vertices of a degenerate vertex of a convex polytope. In the following we call it the neighbourhood problem. The determination of all neighbouring vertices of a nondegenerate vertex is unproblematic, since a nondegenerate vertex has exactly one pivot tableau and all neighbouring vertices can be determined on the basis of this tableau in one pivot step each time. In contrast to this, a degenerate vertex x^o is associated with different pivot tableaux, and in exceptional cases only can all neighbouring vertices be computed using just one pivot tableau of x^{o 1)}.