Victor Klee

THE D-STEP CONJECTURE FOR POLYHEDRA OF DIMENSION D<6,

Two functions Δ and Δb, of interest in combinatorial geometry and the theory of linear programming, are defined and studied. Δ(d, n) is the maximum diameter of convex polyhedra of dimensiond withn faces of dimensiond−1; similarly, Δb(d,n) is the maximum diameter of bounded polyhedra of dimensiond withn faces of dimensiond−1. The diameter of a polyhedronP is the smallest integerl such that any two vertices ofP can be joined by a path ofl or fewer edges ofP. It is shown that the boundedd-step conjecture, i.e. Δb(d,2d)=d, is true ford≤5. It is also shown that the generald-step conjecture, i.e. Δ(d, 2d)≤d, of significance in linear programming, is false ford≥4. A number of other specific values and bounds for Δ and Δb are presented.

The d-Step Conjecture and Its Relatives

The d-step conjecture arose from an attempt to understand the computational complexity of edge-following algorithms for linear programming, such as the simplex algorithm. It can be stated in terms of diameters of graphs of convex polytopes, in terms of the existence of nonrevisiting paths in such graphs, in terms of an exchange process for simplicial bases of a vector space, and in terms of matrix pivot operations. First formulated by W. M. Hirsch in 1957, the conjecture remains unsettled, though it has been proved in many special cases and counterexamples have been found for slightly stronger conjectures. If the conjecture is false, as we believe to be the case, then finding a counterexample will be merely a small first step in the line of investigation related to the conjecture. This report summarizes what is known about the d-step conjecture and its relatives. A considerable amount of new material is included, but it does not seem to come close to settling the conjecture. Of special interest is the first example of a polytope that is not vertex-decomposable, showing that a certain natural approach to the conjecture will not work. Also significant are the quantitative relations among the lengths of paths associated with various forms of the conjecture.

ON THE NUMBER OF VERTICES OF A CONVEX POLYTOPE

Abstract : Contents: Eulerian manifolds; Polytopes and pyramids; The number of vertices of a convex polytope; The problem of Dantzig.

Many Polytopes Meeting the Conjectured Hirsch Bound

Abstract. The still open Hirsch conjecture asserts that Δ(d,n) ≤n-d for all n > d ≥ 2 , where Δ (d,n) denotes the maximum edge-diameter of (convex) d -polytopes with n facets. This paper adds to the list of pairs (d,n) that are known to be H -sharp in the sense that Δ (d,n) ≥ n-d . In particular, it is proved that Δ(d,n)≥ n-d for all n > d ≥ 14 .

Mathematical Programming and Convex Geometry

Publisher Summary This chapter discusses parts of convex geometry that are important for optimization in mathematical programming. Mathematical programming consists of methods for finding or approximating the global optimum of a real-valued function φ over a domain D in real n -space ℝ d or over D ⋂ℚ n or D ⋂ℤ n (the rational or integer points in D ). Linear programming is generally understood to imply that the objective function φ is linear, that the domain D is a convex polyhedron given as the intersection of a finite number of closed half spaces, and that the optimization is over D or D ⋂ℚ n . The chapter discusses some important and contrasting aspects of convex minimization and convex maximization—(1) relationship between local and global extrema; (2) extent to which optimization is an interior- or boundary-point phenomenon; (3) extent to which optimization reduces to one-dimensional considerations; and (4) complexity of quadratic optimization over polytopes. If φ is an affine function, then φ is convex, so the problem of minimizing φ has some aspects in common with convex minimization. Similarly, if φ is concave, then the problem of minimizing φ has aspects in common with concave minimization, or, equivalently, with convex maximization.

A class of linear programming problems requiring a large number of iterations

Abstract : A coordinate-free description of the simplex algorithm (for nondegenerate linear programming problems) is supplied, and is used to show that the number of iterations can be larger than was previously known. For 0 m n, there is constructed a nondegenerate linear programming problem whose bounded (n - m)-dimensional feasible region is defined by means of m linear equality constraints in n nonnegative variables, and in which, after starting from the worst choice of an initial feasible vertex, m(n - m - l) + 1 simplex iterations are required in order to reach the optimal vertex. It is conjectured that this is the maximum possible number of iterations (for arbitrary 0 m n), but the conjecture is proved only for n m + 4. (Author)

HEIGHTS OF CONVEX POLYTOPES

Abstract : Though phrased in geometric terms, this report is concerned with the comparative efficiency of various pivot rules for the simplex method of linear programming, in which one seeks to maximize a linear objective function by moving along the edges of the feasible region. The following three rules are considered: (1) from the vertex v, survey the adjacent vertices until a vertex w is found for which the linear function of v is less than that for w, then move to w and continue the process; (2) from the vertex v, survey all of the adjacent vertices in order to find one, say w, at which the linear function has the greatest value; then move to w and continue the process; (3) from the vertex v, survey all of the adjacent vertices in order to find one, say w, for which the slope has the greatest value; then move to w and continue the process. (Author)