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Linear Complementarity and Oriented Matroids
 Journal of the Operational Research Society of Japan
, 1990
"... 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 t ..."
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Cited by 12 (8 self)
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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 crisscross method which solves one of the primal or dual problem by using surprisingly simple pivot rules (without any pertur...
A Survey on Pivot Rules for Linear Programming
 ANNALS OF OPERATIONS RESEARCH. (SUBMITTED
, 1991
"... 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. Th ..."
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Cited by 9 (1 self)
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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...
The Linear Complementarity Problem, Sufficient Matrices and the CrissCross Method
, 1990
"... Specially structured Linear Complementarity Problems (LCP's) and their solution by the crisscross method are examined in this paper. The crisscross method is known to be finite for LCP's with positive semidefinite bisymmetric matrices and with Pmatrices. It is also a simple finite algorithm for o ..."
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Cited by 6 (4 self)
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Specially structured Linear Complementarity Problems (LCP's) and their solution by the crisscross method are examined in this paper. The crisscross method is known to be finite for LCP's with positive semidefinite bisymmetric matrices and with Pmatrices. It is also a simple finite algorithm for oriented matroid programming problems. Recently Cottle, Pang and Venkateswaran identified the class of (column, row) sufficient matrices. They showed that sufficient matrices are a common generalization of P and PSDmatrices. Cottle also showed that the principal pivoting method (with a clever modification) can be applied to row sufficient LCP's. In this paper the finiteness of the crisscross method for sufficient LCP's is proved. Further it is shown that a matrix is sufficient if and only if the crisscross method processes all the LCP's defined by this matrix and all the LCP's defined by the transpose of this matrix and any parameter vector.
Principal Pivoting Methods For Linear Complementarity Problems, PCPLCP
"... 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 diago ..."
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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