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# Lemke's algorithm

About: Lemke's algorithm is a research topic. Over the lifetime, 303 publications have been published within this topic receiving 12384 citations.

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18 Feb 1992

TL;DR: In this article, the authors present an overview of existing and multiplicity of degree theory and propose pivoting methods and iterative methods for degree analysis, including sensitivity and stability analysis.

Abstract: Introduction. Background. Existence and Multiplicity. Pivoting Methods. Iterative Methods. Geometry and Degree Theory. Sensitivity and Stability Analysis. Chapter Notes and References. Bibliography. Index.

2,837 citations

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TL;DR: An algebraic proof of the existence of equilibrium points for two-person non-zero-sum games is given in this paper, leading to an efficient scheme for computing an equilibrium point, which is valid for any ordered field.

Abstract: An algebraic proof is given of the existence of equilibrium points for bimatrix (or two-person, non-zero-sum) games. The proof is constructive, leading to an efficient scheme for computing an equilibrium point. In a nondegenerate case, the number of equilibrium points is finite and odd. The proof is valid for any ordered field.

1,036 citations

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TL;DR: In this paper, simple constructive proofs are given of solutions to the matric matric system Mz − ω = q; z ≧ 0; ω ≧ 1; zT = 0, for various kinds of data M, q, which embrace quadratic programming and the problem of finding equilibrium points of bimatrix games.

Abstract: Some simple constructive proofs are given of solutions to the matric system Mz − ω = q; z ≧ 0; ω ≧ 0; and zT ω = 0, for various kinds of data M, q, which embrace the quadratic programming problem and the problem of finding equilibrium points of bimatrix games. The general scheme is, assuming non-degeneracy, to generate an adjacent extreme point path leading to a solution. The scheme does not require that some functional be reduced.

948 citations

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TL;DR: The role of problems of the form w and z satisfying w = q + Mz, w = or 0, z = or0, zw = 0 play a fundamental role in mathematical programming.

Abstract: : Problems of the form: Find w and z satisfying w = q + Mz, w = or 0, z = or 0, zw = 0 play a fundamental role in mathematical programming. This paper describes the role of such problems in linear programming, quadratic programming and bimatrix game theory and reviews the computational procedures of Lemke and Howson, Lemke, and Dantzig and Cottle.

728 citations