this paper, we show how one can design new pivot algorithms for solving the LP and the LCP. In particular, we are interested in combinatorial pivot algorithms which solve the LP and a certain class of LCP's. Here, a pivot algorithm is called combinatorial if the pivot choice depends only on the signs of entries of their dictionaries. The best source of combinatorial pivot algorithms is in the theory of oriented matroid (OM) programming [Bla77a, Edm94, Fuk82, FT92, LL86, Ter87, Tod85, Wan87]. The well-known Bland's pivot rule [Bla77b] for the simplex method can be considered as a combinatorial algorithm, but it is not a typical one. The main characteristic of the "OM" algorithms is that the feasibility may not be preserved at all in both primal and dual problem, and the finiteness of the algorithms is guaranteed by some purely combinatorial improvement argument rather than by the reasoning based on the increment of the objective function value. One immediate advantage of combinatorial algorithms is that the degeneracy does not have to be treated separately. Thus a very simple combinatorial algorithm, such as the criss-cross method [Ter87, Wan87], solves the general LP correctly and yields one of the simplest proofs of the strong duality theorem. There is a well-noted disadvantage of combinatorial algorithms. The number of pivot operations to solve the LP tends to grow rapidly in practice. Furthermore it is often quite easy to construct a class of LP's for which a given combinatorial algorithm takes an exponential number of pivot operations in the input size. In this paper, we review the finiteness proof of combinatorial algorithms and study a new algorithm in the class. The key ingredients of the new algorithm are "history dependency" and "largest combinatorial improveme...

Abstract not found

In [Man82] A. Mandel proved that the maximal cells of an Oriented Matroid poset are B-shellable. Our result shows that the whole Oriented Matroid is shellable, too.

Point configurations and convex polytopes play central roles in computational geometry and discrete geometry. For many problems, their combinatorial structures, i.e., the underlying oriented matroids up to isomorphism,

A general framework of nite algorithms is presented here for quadratic programming. This algorithm is a direct generalization of Van der Heyden's algorithm for the linear complementarity problem and Jensen's `relaxed recursive algorithm', which was proposed for solution of Oriented Matroid programming problems. The validity of this algorithm is proved the same way as the finiteness of the criss-cross method is proved. The second part of this paper contains a generalization of Edmonds-Fukuda pivoting rule for quadratic programming. This generalization can be considered as a finite version of Van de Panne - Whinston algorithm and so it is a simplex method for quadratic programming. These algorithms uses general combinatorial type ideas, so the same methods can be applied for oriented matroids as well. The generalization of these methods for oriented matroids is a subject of another paper.

. Assuming that the reader is familiar with both the primal and dual simplex methods, Zionts' criss-cross method can easily be explained. ffl It can be initialized by any, possibly both primal and dual infeasible basis . If the basis is optimal, we are done. If the basis is not optimal , then there are some primal or dual infeasible variables. One might choose any of these. It is advised to choose once a primal and then a dual infeasible variable, if possible. ffl If the selected variable is dual infeasible, then it enters the basis and the leaving variable is chosen among the primal feasible variables in such a way that primal feasibility of the currently primal feasible variables is preserved. If no such basis exchange is possible another infeasible variable is selected. ffl If the selected variable is primal infeasible, then it leaves the basis and the entering variable is chosen among th

this paper. leastindex rules were designed for network flow problems, linear optimization problems, linear complementarity problems and oriented matroid programming problems. These classes will be considered in the sequel.

this process is repeated endlessly. Because the simplex method produces a sequence with monotonically improving objective values, the objective stays constant in a cycle, thus each pivot in the cycle must be degenerate. The possibility of cycling was recognized shortly after the invention of the simplex algorithm. Cycling examples were given by E.M.L. Beale [2] and by A.J. Hoffman [10]. Recently a scheme to construct cycling LO examples is presented in [9]. These examples made evident that extra techniques are needed to ensure finite termination of simplex methods. The first and widely used such tool is the lexicographic simplex rule. Other techniques, like the leastindex anti-cycling rules and more general recursive schemes were developed more recently. Lexicographic simplex methods. First we need to define an ordering, the so-called lexicographic ordering of vectors. Lexicograph

We study the space of all extensions of a real hyperplane arrangement by a new pseudo-hyperplane, and, more generally, of an oriented matroid by a new element. The question whether this space has the homotopy type of a sphere is a special case of the "Generalized Baues Problem" of Billera, Kapranov & Sturmfels, via the Bohne-Dress Theorem on zonotopal tilings. We prove that the extension space is spherical for the class of strongly euclidean oriented matroids. This class includes the alternating matroids and all oriented matroids of rank at most 3 or of corank at most 2. In general it is not even known whether the extension space is connected. We show that the subspace of realizable extensions is always connected but not necessarily spherical.