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The Linear Complementarity Problem

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  • B. Curtis Eaves

    (University of California, Berkeley)

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    Abstract

    This study centers on the task of efficiently finding a solution of the linear complementarity problem: Ix - My = q, x \ge 0, y \ge 0, x \perp y. The main results are: (1) It is shown that Lemke's algorithm will solve (or show no solution exists) the problem for M \in L where L is a class of matrices, which properly includes (i) certain copositive matrices, (ii) certain matrices with nonnegative principal minors, (iii) matrices for bimatrix games. (2) If M \in L, if the system Ix - My = q, x \ge 0, y \ge 0 is feasible and nondegenerate, then the corresponding linear complementarity problem has an odd number of solutions. If M \in L and q > 0 then the solution is unique. (3) If for some M and every q \ge 0 the problem has a unique solution then M \in L and the problem has a solution for every q. (4) If M has nonnegative principal minors and if the linear complementarity with M and q has a nondegenerate complementary solution then the solution is unique. (5) If y TMy + y Tq is bounded below on y \ge 0 then the linear complementarity problem with M and q has a solution and Lemke's algorithm can be used to find such a solution. If, in addition, the problem is nondegenerate, then it has an odd number of solutions. (6) A procedure based on Lemke's algorithm is developed which either computes stationary points for general quadratic programs or else shows that the program has no optimum. (7) If a quadratic program has an optimum and satisfies a nondegeneracy condition then there are an odd number of stationary points.

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    Bibliographic Info

    Article provided by INFORMS in its journal Management Science.

    Volume (Year): 17 (1971)
    Issue (Month): 9 (May)
    Pages: 612-634

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    Handle: RePEc:inm:ormnsc:v:17:y:1971:i:9:p:612-634

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    Cited by:
    1. Srihari Govindan & Robert Wilson, 2010. "A decomposition algorithm for N-player games," Economic Theory, Springer, vol. 42(1), pages 97-117, January.
    2. Anne Balthasar, 2010. "Equilibrium tracing in strategic-form games," Economic Theory, Springer, vol. 42(1), pages 39-54, January.
    3. Christian Bidard & Guido Erreygers, 1998. "The number and type of long-term equilibria," Journal of Economics, Springer, vol. 67(2), pages 181-205, June.
    4. Thomas Quint & Martin Shubik, 1994. "On the Number of Nash Equilibria in a Bimatrix Game," Cowles Foundation Discussion Papers 1089, Cowles Foundation for Research in Economics, Yale University.
    5. Herings, P.J.J. & Elzen, A.H. van den, 1998. "Computation of the Nash Equilibrium Selected by the Tracing Procedure in N-Person Games," Discussion Paper 1998-04, Tilburg University, Center for Economic Research.
    6. Shu-Cherng Fang & Elmor L. Peterson, 1979. "A Unification and Generalization of the Eaves and Kojima Fixed Point Representations of the Complementarity Problem," Discussion Papers 365, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
    7. Richard Cottle, 2010. "A field guide to the matrix classes found in the literature of the linear complementarity problem," Journal of Global Optimization, Springer, vol. 46(4), pages 571-580, April.
    8. Govindan, Srihari & Wilson, Robert, 2003. "A global Newton method to compute Nash equilibria," Journal of Economic Theory, Elsevier, vol. 110(1), pages 65-86, May.
    9. Thomas Quint & Martin Shubik, 1995. "A Bound on the Number of Nash Equilibria in a Coordination Game," Cowles Foundation Discussion Papers 1095, Cowles Foundation for Research in Economics, Yale University.
    10. Gailly, B. & Installe, M. & Smeers, Y., 2001. "A new resolution method for the parametric linear complementarity problem," European Journal of Operational Research, Elsevier, vol. 128(3), pages 639-646, February.
    11. Bernhard Stengel, 2010. "Computation of Nash equilibria in finite games: introduction to the symposium," Economic Theory, Springer, vol. 42(1), pages 1-7, January.
    12. Herings, P.J.J. & Laan, G. van der & Talman, A.J.J., 1998. "Price-Quantity Adjustment in a Keynesian Economy," Discussion Paper 1998-118, Tilburg University, Center for Economic Research.

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