# Introduction

• H.-J. Kruse
Chapter
Part of the Lecture Notes in Economics and Mathematical Systems book series (LNE, volume 260)

## Abstract

There are many mathematical (optimization) models all feasible solutions of which represent a convex polyhedral set or especially a convex polytope1). For determining the vertices of convex polytopes we have problem-oriented pivoting methods. The application of these methods in case of degeneracy, i.e. the convex polytope contains degenerate (overdetermined) vertices, involves various difficulties. Efficiency problems or convergence problems will arise2). This is due to the fact that a great number of bases or pivot tableaux is associated with a degenerate vertex (cf. Chapter 4). The ratio e : b of number of vertices e and number of bases b of a convex polytope is the smaller, the greater the proportion of degenerate vertices and the more extreme the degeneracy of these vertices is3). Since the pivoting methods usually generate many (often even all) pivot tableaux for a degenerate vertex, the computational effort substantially depends upon the number and the individual degeneracy degree of the degenerate vertices which are (must be) determined with the help of such techniques. Therefore efficiency losses due to degeneracy have to be taken into account, especially in using pivoting methods suited for determining all vertices of convex polytopes1). However, in the.literature dealing with these methods there are no special approaches allowing an efficient procedure in case of degeneracy2).