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Complementarity problems

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Encyclopedia of Operations Research and Management Science

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DEFINITION

In its most elementary form, a complementarity problem CP(f) is an inequality system stated in terms of a mapping f: R n → R n. Given f, one seeks a vector x ∈ R n such that

(1)

When the mapping f is affine, say of the form f (x) = q + Mx, the problem (1) is called a linear complementarity problem, denoted LCP(q,M) or sometimes just (q,M). Otherwise, it is called a nonlinear complementarity problem and is denoted NCP(f).

If is a solution to (1) satisfying the additional nondegeneracy condition i, ..., n, the indices i for which i form complementary subsets of {1, ..., n}. This is believed to be the origin of the term complementary slackness as used in linear and nonlinear programming. It was this terminology that inspired the name complementarity problem.

SOURCES OF COMPLEMENTARITY PROBLEMS

The complementarity problem is intimately linked to the Karush-Kuhn-Tucker necessary conditions of local optimality found in mathematical programming theory. This connection was brought...

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References

  1. Cottle, R. W. (1964). Nonlinear Programs with Positively Bounded Jacobians, Ph.D. Thesis, Department of Mathematics, University of California, Berkeley. [See also, Technical Report ORC 64-12 (RR), Operations Research Center, University of California, Berkeley.]

    Google Scholar 

  2. Cottle, R. W. (1966). “Nonlinear Programs with Positively Bounded Jacobians,” SIAM Jl. Applied Mathematics 14, 147–158.

    Google Scholar 

  3. Cottle, R. W. and Dantzig, G. B. (1968). “Complementary Pivot Theory of Mathematical Programming, Linear Algebra and its Applications 1, 103–125.

    Google Scholar 

  4. Cottle, R. W. and Dantzig, G. B. (1970). “A Generalization of the Linear Complementarity Problem, Jl. Combinatorial Theory 8, 79–90.

    Google Scholar 

  5. Cottle, R. W., Pang, J. S., and Stone, R. E. (1992). The Linear Complementarity Problem, Academic Press, Boston.

    Google Scholar 

  6. De Schutter, B. and De Moor, B. (1996). “The Extended Linear Complementarity Problem,” Mathematical Programming 71, 289–326.

    Google Scholar 

  7. Eaves, B. C. (1971). “The Linear Complementarity Problem,” Management Science 17, 612–634.

    Google Scholar 

  8. Eaves, B. C. and Lemke, C. E. (1981). “Equivalence of LCP and PLS,” Mathematics of Operations Research 6, 475–484.

    Google Scholar 

  9. Ebiefung, A. A. (1995). “Existence Theory and Q-matrix Characterization for the Generalized Linear Complementarity Problem,” Linear Algebra and Its Applications 223/224, 155–169.

    Google Scholar 

  10. Ferris, M. C., and Pang, J. S. “Engineering and Economic Applications of Complementarity Problems,” SIAM Review 39, 669–713.

    Google Scholar 

  11. Gowda, M. S. (1995). “On Reducing a Monotone Horizontal LCP to an LCP,” Applied Mathematics Letters 8, 97–100.

    Google Scholar 

  12. Gowda, M. S. (1996). “On the Extended Linear Complementarity Problem,” Mathematical Programming 72, 33–50.

    Google Scholar 

  13. Habetler, G. J. and Price, A. J. (1971). “Existence Theory for Generalized Nonlinear Complementarity Problems,” Jl. Optimization Theory and Applications 7, 223–239.

    Google Scholar 

  14. Harker, P. T. and Pang, J. S. (1990). “Finite-Dimensional Variational Inequality and Nonlinear Complementarity Problems: A Survey of Theory, Algorithms and Applications,” Mathematical Programming, Series B 48, 161–220.

    Google Scholar 

  15. Howson, J. T., Jr. (1963). Orthogonality in Linear Systems, Ph.D. Thesis, Department of Mathematics, Rensselaer Institute of Technology, Troy, New York.

    Google Scholar 

  16. Isac, George (1992). Complementarity Problems, Lecture Notes in Mathematics 1528, Springer–Verlag, Berlin.

    Google Scholar 

  17. Karamardian, S. (1971). “Generalized Complementarity Problem,” Jl. Optimization Theory and Applications 8, 161–168.

    Google Scholar 

  18. Kojima, M., Megiddo, N., Noma, T., and Yoshise, A. (1991). A Unified Approach to Interior Point Algorithms for Linear Complementarity Problems, Lecture Notes in Computer Science 538, Springer-Verlag, Berlin.

    Google Scholar 

  19. Lemke, C. E. (1965). “Bimatrix Equilibrium Points and Mathematical Programming,” Management Science 11, 681–689.

    Google Scholar 

  20. Lemke, C. E. and Howson, J. T., Jr. (1964). “Equilibrium Points of Bimatrix Games,” SIAM Jl. Applied Mathematics 12, 413–423.

    Google Scholar 

  21. Mangasarian, O. L., and Pang, J. S. (1995). “The Extended Linear Complementarity Problem,” SIAM Jl. Matrix Analysis and Applications 16, 359–368.

    Google Scholar 

  22. Mangasarian, O. L., and Solodov, M. V. (1993). “Nonlinear Complementarity as Unconstrained and Constrained Minimization,” Mathematical Programming, Series B 62, 277–297.

    Google Scholar 

  23. Mohan, S. R., and Neogy, S. K. (1997). “Vertical Block Hidden Z-matrices and the Generalized Linear Complementarity Problem,” SIAM Jl. Matrix Analysis and Applications 18, 181–190.

    Google Scholar 

  24. Murty, K. G. (1968). On the Number of Solutions to the Complementarity Problem and Spanning Properties of Complementary Cones, Ph.D. Thesis, Department of Industrial Engineering and Operations Research, University of California, Berkeley.

    Google Scholar 

  25. Murty, K. G. (1972). “On the Number of Solutions to the Complementarity Problem and Spanning Properties of Complementary Cones,” Linear Algebra and Its Applications 5, 65–108.

    Google Scholar 

  26. Murty, K. G. (1988). Linear Complementarity, Linear and Nonlinear Programming, Heldermann-Verlag, Berlin.

    Google Scholar 

  27. Pang, J. S. (1995). “Complementarity Problems,” in Handbook of Global Optimization, R. Horst and P. Pardalos, eds., Kluwer, Dordrecht, 271–338.

    Google Scholar 

  28. Robinson, S. M. (1979). “Generalized Equations and Their Solutions, Part I: Basic Theory,” Mathematical Programming, Study 10, 128–141.

    Google Scholar 

  29. Samelson, H., Thrall, R. M., and Wesler, O. (1958). “A Partition Theorem for Euclidean n-Space,” Proceedings American Mathematical Society 9, 805–807.

    Google Scholar 

  30. Sznajder, R. and Gowda, M. S. (1995). “Generalizations of P 0- and P-properties; Extended Vertical and Horizontal LCPs,” Linear Algebra and Its Applications 223/224, 695–715.

    Google Scholar 

  31. Tütüncü, R. H., and Todd, M. J. (1995). “Reducing Horizontal Linear Complementarity Problems,” Linear Algebra and Its Applications 223/224, 717–730.

    Google Scholar 

  32. Zhang, Y. (1994). “On the Convergence of a Class of Infeasible Interior-point Algorithm for the Horizontal Linear Complementarity Problem,” SIAM Jl. Optimization 4, 208–227.

    Google Scholar 

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Authors and Affiliations

  1. Stanford University, Stanford, California, USA

    Richard W. Cottle

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  1. Richard W. Cottle
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Editors and Affiliations

  1. Robert H. Smith School of Business, University of Maryland, College Part, Maryland, USA

    Saul I. Gass

  2. School of Information Technology & Engineering, George Mason University, Fairfax, Virginia, USA

    Carl M. Harris

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© 2001 Kluwer Academic Publishers

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Cottle, R.W. (2001). Complementarity problems . In: Gass, S.I., Harris, C.M. (eds) Encyclopedia of Operations Research and Management Science. Springer, New York, NY. https://doi.org/10.1007/1-4020-0611-X_139

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  • DOI: https://doi.org/10.1007/1-4020-0611-X_139

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  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-0-7923-7827-3

  • Online ISBN: 978-1-4020-0611-1

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