Bounds on the number of vertices of perturbed polyhedra

Degeneracy of a polyhedron is a source of difficulties for both theory and computation in mathematical programming. This phenomenon may be avoided by considering slight perturbations of degenerate polyhedra, that is, by approximating the degenerate polyhedron by a nondegenerate one. This method was applied to a well-known proof of finiteness of the simplex algorithm [5, 7]. Another application is a connectedness result of Gal [9-11, 15] concerning graphs of bases associated to the degenerate vertices of a polyhedron, which had induced an algorithm for finding all incident edges to a degenerate vertex [20]. So the pivoting methods which require the search of all or part of the incident edges to a given vertex, for example the vertex enumeration methods (see for example [3, 21] and a survey in [8]), the linear multiobjective methods (see for example [1] and references given therein), frequently need a perturbation argument such that the polyhedron is simple. But the perturbed polyhedron has generally many more vertices than the initial polyhedron. Here we give one lower and two upper bounds on the number of vertices of perturbed polyhedra. In our study, we resolutely adopt a geometric viewpoint. This permits the use of some basic facts in the field of convex polytopes as well as a strong result of Klee [19] concerning the number of vertices of unbounded polyhedra (see also [2]). This paper is organized as follows. In section 2 we extend the right hand side perturbation principle by introducting a perturbation function of a polyhedron and we define the corresponding perturbed polyhedron. We show that, for a given function, the family of perturbed polyhedra belongs to an equivalence class under the