this paper is to present mathematical ideas and

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...

We discuss methods for the generation of oriented matroids and of isomorphism classes of oriented matroids. Our methods are based on single element extensions and graph theoretical representations of oriented matroids, and all these methods work in general rank and for nonuniform and uniform oriented matroids as well. We consider two types of graphs, cocircuit graphs and tope graphs, and discuss the single element extensions in terms of localizations which can be viewed as partitions of the vertex sets of the graphs. Whereas localizations of the cocircuit graph are well characterized, there is no graph theoretical characterization known for localizations of the tope graph. In this paper we prove a connectedness property for tope graph localizations and use this for the design of algorithms for the generation of single element extensions by use of tope graphs. Furthermore we discuss similar algorithms which use the cocircuit graph. The characterization of localizations of cocircuit graphs nally leads to a backtracking algorithm which is a simple and efficient method for the generation of single element extensions. We compare this method with a recent algorithm of Bokowski and Guedes de Oliveira for uniform oriented matroids.

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...

. In this short paper, we prove the finiteness of the criss-cross method by showing a certain binary number of bounded digits associated with each iteration increases monotonically. This new proof immediately suggests the possibility of relaxing the pivoting selection in the criss-cross method without sacrificing the finiteness. Key Words: linear programming. simplex method, finite pivoting rules. 1 The Criss-Cross Method Let A be an m2 n matrix. Let E be the index set of columns of the matrix A; and f; g be two distinct members of E: Here we consider the standard form linear program: (P ) maximize x f (1.1) subject to A x = 0; (1.2) x g = 1; (1.3) x j 0; 8 j 2 E 0 ff; gg: (1.4) A vector x is said to be feasible if it satisfies the constraints (1.2), (1.3), and (1.4). If a linear program has a feasible solution, then it is called feasible, otherwise it is called infeasible. For any linear program, we will refer to following three situations as characters: 3 Supported by Grant...

this paper we have given an alternative proof of Farkas's lemma, a proof that is based on a theorem, the main theorem, that relates to the eigenvectors of certain orthogonal matrices. This theorem is believed to be new, and the author is not aware of any similar theorem concerning orthogonal matrices although he recently proved the weak form of the theorem using Tucker's theorem (see [5]). His proof of the theorem is "completely elementary" (a referee) and requires little more than a knowledge of matrix algebra for its understanding. Once the theorem is established, Tucker's theorem (via the Cayley transform), Farkas's lemma and many other theorems of the alternative follow trivially. Thus the paper establishes a connection between the eigenvectors of orthogonal matrices, duality in linear programming and theorems of the alternative that is not generally appreciated, and this may be of some theoretical interest.

Abstract. This paper deals with a geometric construction of algebraic non-realizability proofs for certain oriented matroids. As main result we obtain an algorithm which generates a (bi-quadratic) final polynomial [3], [5] for any non-euclidean oriented matroid. Here we apply the results of Edmonds, Fukuda and Mandel [6], [7] concerning non-degenerate cycling of linear programms in non-euclidean oriented matroids. 1.

this paper we present a unified framework in which we describe two known algorithms as special simplex methods and analyse their complexities and differences

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...