Some generalizations of the criss-cross method for the linear complementarity problem of oriented matroids

Abstract

Quadratic programming, symmetry, positive (semi) definiteness and the linear complementary problem were generalized by Morris and Todd to oriented matroids. Todd gave a constructive solution for the quadratic programming problem of oriented matroids. Using Las Vergnas' lexicographic extension and Bland's basic tableau construction Todd generalized Lemke's quadratic programming algorithm for this problem.

Here some generalizations of Terlaky's finite criss-cross method are presented for oriented matroid quadratic programming. These algorithms are based on the smallest subscript rule and on sign patterns, and do not preserve feasibility on any subsets. In fact two variants of the generalized criss-cross method are presented. Finally two special cases (oriented matroid linear programming and the definite case) are discussed.

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Klafszky, E., Terlaky, T. Some generalizations of the criss-cross method for the linear complementarity problem of oriented matroids. Combinatorica 9, 189–198 (1989). https://doi.org/10.1007/BF02124679

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