Three generalizations of the criss-cross method for quadratic programming are presented here. Tucker's, Cottle's and Dantzig's principal pivoting methods are specialized as diagonal and exchange pivots for the linear complementarity problem obtained from a convex quadratic program. A finite criss-cross method, based on least-index resolution, is constructed for solving the LCP. In proving finiteness, orthogonality properties of pivot tableaus and positive semidefiniteness of quadratic matrices are used. In the last section some special cases and two further variants of the quadratic criss-cross method are discussed. If the matrix of the LCP has full rank, then a surprisingly simple algorithm follows, which coincides with Murty's `Bard type schema' in the P matrix case.

A combinatorial abstraction of the linear complementarity theory in the setting of oriented matroids was rst considered by M.J. Todd. In this paper, we take a fresh look at this abstraction, and attempt to give a simple treatment of the combinatorial theory of linear complementarity. We obtain new theorems, proofs and algorithms in oriented matroids whose specializations to the linear case are also new. For this, the notion of suciency of square matrices, introduced by Cottle, Pang and Venkateswaran, is extended to oriented matroids. Then, we prove a sort of duality theorem for oriented matroids, which roughly states: exactly one of the primal and the dual system has a complementary solution if the associated oriented matroid satisfies "weak" sufficiency. We give two different proofs for this theorem, an elementary inductive proof and an algorithmic proof using the criss-cross method which solves one of the primal or dual problem by using surprisingly simple pivot rules (without any pertur...

The purpose of this paper is to survey the various pivot rules of the simplex method or its variants that have been developed in the last two decades, starting from the appearance of the minimal index rule of Bland. We are mainly concerned with the finiteness property of simplex type pivot rules. There are some other important topics in linear programming, e.g. complexity theory or implementations, that are not included in the scope of this paper. We do not discuss ellipsoid methods nor interior point methods. Well known classical results concerning the simplex method are also not particularly discussed in this survey, but the connection between the new methods and the classical ones are discussed if there is any. In this paper we discuss three classes of recently developed pivot rules for linear programming. The first class (the largest one) of the pivot rules we discuss is the class of essentially combinatorial pivot rules. Namely these rules only use labeling and signs of the variab...

We prove that all combinatorial differential manifolds involving only Euclidean oriented matroids are PL manifolds. In doing so we introduce a new notion of triangulations of oriented matroids, and show that any triangulation of a Euclidean oriented matroid is a PL sphere. In Section 5 we adapt these results to get a new definition of triangulations of oriented matroid polytopes, and show that any triangulation of a Euclidean oriented matroid polytope is a PL ball. 1 Introduction In [M], MacPherson introduced a new class of combinatorial objects called combinatorial differential manifolds, or CD manifolds. The idea of a CD manifold is to give a simplicial complex a combinatorial analog to a differential structure. The role of "tangent spaces" in this theory is played by oriented matroids, objects about which we will have much more to say later. If M is a real differential manifold and j : jjXjj !M is a smooth triangulation of M , then we shall see in Section 2.2 that the differentia...

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

timization problem min ae c T x + 1 2 x T Qx : Ax b; x 0 oe ; where Q is a positive semidefinite, symmetric matrix, then M = ` 0 A \GammaA T Q ' and q = ` \Gammab c ' : Here M is a positive semidefinite bisymmetric matrix. Bisymmetry means that the matrix has a block diagonal structure, and it is the sum of a symmetric block diagonal positive semidefinite, and a skew symmetric block diagonal matrix. Some other classes of solvable LCPs are problems, when M is a ffl P -matrix ; ffl sufficient matrix or, equivalently, a P

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

The concept of valuated matroids was introduced by Dress and Wenzel as a quantitative extension of the base exchange axiom for matroids. This paper gives several sets of cryptomorphically equivalent axioms of valuated matroids in terms of (R[f01g)-valued vectors defined on the circuits of the underlying matroid, where R is a totally ordered additive group. The dual of a valuated matroid is characterized by an orthogonality of (R [ f01g)- valued vectors on circuits. Minty's characterization for matroids by the painting property is generalized for valuated matroids.

In this chapter we present the basic mathematical results on the LCP. Many of these results are used in later chapters to develop algorithms to solve LCPs, and to study the computational complexity of these algorithms. Here, unless stated otherwise, I denotes the unit matrix of order n. M is a given square matrix of order n. In tabular form the LCP (q � M) is w z q I;M q w> 0 � z> 0 � w T z =0 (3:1)