Journal Article10.1287/MOOR.1.1.54
Orientation in Complementary Pivot Algorithms
TL;DR: The weakest known general sufficiency conditions are provided for Lemke's algorithm to process the linear complementarity problem and it is shown that fixed point algorithms will only approximate fixed points of f at which the determinant of the Jacobian of f minus the identity has the appropriate sign.
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Abstract: Shapley introduced an index theory for bimatrix games that oriented the paths generated by the Lemke-Howson algorithm and thus partitioned equilibrium points into two sets. Here we develop a similar orientation theory for a generalized complementary pivot algorithm and apply our results to bimatrix games, the linear complementarity problem, and fixed point algorithms. We provide the weakest known general sufficiency conditions (based on those of Garcia) for Lemke's algorithm to process the linear complementarity problem. We also show that if the linear complementarity version of a quadratic programming problem is solved and gives a Kuhn-Tucker solution, then the hessian of the objective function restricted to the subspace of active constraints has positive determinant. Finally we show that fixed point algorithms will only approximate fixed points of f at which the determinant of the Jacobian of f minus the identity has the appropriate sign.
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References
Equilibrium Points of Bimatrix Games
C. E. Lemke,J. T. Howson +1 more
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.
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Bimatrix Equilibrium Points and Mathematical Programming
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.
Complementary pivot theory of mathematical programming
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.
807
•Journal Article
Equilibrium points of bimatrix games
TL;DR: An algebraic proof of the existence of equilibrium points for two-person non-zero-sum games is given in this article, leading to an efficient scheme for computing an equilibrium point, which is valid for any ordered field.
785
The Linear Complementarity Problem
TL;DR: In this article, it was shown that Lemke's algorithm will not solve the linear complementarity problem for a class of matrices, which properly includes copositive matrices with nonnegative principal minors, and matrices for bimatrix games.
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