On the representation of polyhedra by polynomial inequalities

A beautiful result of Br\"ocker and Scheiderer on the stability index of basic closed semi-algebraic sets implies, as a very special case, that every $d$-dimensional polyhedron admits a representation as the set of solutions of at most $d(d+1)/2$ polynomial inequalities. Even in this polyhedral case, however, no constructive proof is known, even if the quadratic upper bound is replaced by any bound depending only on the dimension. Here we give, for simple polytopes, an explicit construction of polynomials describing such a polytope. The number of used polynomials is exponential in the dimension, but in the 2- and 3-dimensional case we get the expected number $d(d+1)/2$.Comment: 19 pages, 4 figures; revised version with minor changes proposed by the referee

175 Years of linear programming: 2. Pivots in column space

The simplex method has been the veritable workhorse of linear programming for five decades now. An elegant geometric interpretation of the simplex method can be visualised by viewing the animation of the algorithm in acolumn space representation. In fact, it is this interpretation that explains why it is called the simplex method. The extreme points of the feasible region (polyhedron) of the linear programme can be shown to correspond to an arrangement of simplices in this geometry and the pivoting operation to a physical pivot from one simplex to an adjacent one in the arrangement. This paper introduces this vivid description of the simplex method as a tutored dance of simplices performing 'pivots in column space'

A cutting-plane approach to the edge-weighted maximal clique problem

We investigated the computational performance of a cutting-plane algorithm for the problem of determining a maximal subclique in an edge-weighted complete graph. Our numerical results are contrasted with reports on closely related problems for which cutting-plane approaches perform well in instances of moderate size. Somewhat surprisingly, we find that our approach already in the case of n = 15 or N = 25 nodes in the underlying graph typically neither produces an integral solution nor yields a good approximation to the true optimal objective function value. This result seems to shed some doubt on the universal applicability of cuttingplane approaches as an efficient means to solve linear (0, 1)-programming problems of moderate size

Geometric Reasoning with polymake

The mathematical software system polymake provides a wide range of functions for convex polytopes, simplicial complexes, and other objects. A large part of this paper is dedicated to a tutorial which exemplifies the usage. Later sections include a survey of research results obtained with the help of polymake so far and a short description of the technical background