Skip to main content

Linear Complementarity Problem

  • Reference work entry

Article Outline

Keywords

Synonyms

Definition

Sources of Linear Complementarity Problems

Equivalent Formulations

The Importance of Matrix Classes

Algorithms for Solving LCPs

Software

Some Generalizations

See also

References

This is a preview of subscription content, log in via an institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   2,500.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD   2,499.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

References

  1. Cottle RW (1964) Nonlinear programs with positively bounded Jacobians. Univ. Calif., Berkeley, CA

    Google Scholar 

  2. Cottle RW (1966) Nonlinear programs with positively bounded Jacobians. SIAM J Appl Math 14:147–158

    Article  MathSciNet  MATH  Google Scholar 

  3. Cottle RW, Dantzig GB (1968) Complementary pivot theory of mathematical programming. Linear Alg & Its Appl 1:103–125

    Article  MathSciNet  MATH  Google Scholar 

  4. Cottle RW, Dantzig GB (1970) A generalization of the linear complementarity problem. J Combin Th 8:79–90

    Article  MathSciNet  MATH  Google Scholar 

  5. Cottle RW, Pang JS, Stone RE (1992) The linear complementarity problem. Acad. Press, New York

    MATH  Google Scholar 

  6. Cottle RW, Pang JS, Venkateswaran V (1989) Sufficient matrices and the linear complementarity problem. Linear Alg & Its Appl 114/115:231–249

    Article  MathSciNet  Google Scholar 

  7. Eaves BC (1971) The linear complementarity problem. Managem Sci 17:612–634

    MathSciNet  MATH  Google Scholar 

  8. Eaves BC, Lemke CE (1981) Equivalence of LCP and PLS. Math Oper Res 6:475–484

    MathSciNet  MATH  Google Scholar 

  9. Ebiefung AA (1995) Existence theory and Q-matrix characterization for generalized linear complementarity problem. Linear Alg & Its Appl 223/224:155–169

    Article  MathSciNet  Google Scholar 

  10. Ferris MC, Pang JS (1997) Engineering and economic applications of complementarity problems. SIAM Rev 39:669–713

    Article  MathSciNet  MATH  Google Scholar 

  11. Gowda MS (1995) On reducing a monotone horizontal LCP to an LCP. Appl Math Lett 8:97–100

    Article  MathSciNet  MATH  Google Scholar 

  12. Gowda MS (1996) On the extended linear complementarity problem. Math Program 72:33–50

    MathSciNet  Google Scholar 

  13. Guu S-M, Cottle RW (1995) On a subclass of P0. Linear Alg & Its Appl 223/224:325–335

    Article  MathSciNet  Google Scholar 

  14. Habetler GJ, Price AJ (1971) Existence theory for generalized nonlinear complementarity problems. J Optim Th Appl 7:223–239

    Article  MathSciNet  MATH  Google Scholar 

  15. Harker PT, Pang JS (1990) Finite-dimensional variational inequality and nonlinear complementarity problems: A survey of theory, algorithms and applications. Math Program B 48:161–220

    Article  MathSciNet  MATH  Google Scholar 

  16. Howson JT Jr (1963) Orthogonality in linear systems. Rensselaer Inst. Techn., Troy, NY

    Google Scholar 

  17. Isac G (1992) Complementarity problems. Lecture Notes Math, vol 1528. Springer, Berlin

    Book  MATH  Google Scholar 

  18. Karamardian S (1971) Generalized complementarity problem. J Optim Th Appl 8:161–168

    Article  MathSciNet  MATH  Google Scholar 

  19. Kojima M, Megiddo N, Noma T, Yoshise A (1991) A unified approach to interior point algorithms for linear complementarity problems. Lecture Notes Computer Sci, vol 538. Springer, Berlin

    Google Scholar 

  20. Kuhn D, Löwen R (1987) Piecewise affine bijections of R n and the equation Sx +Tx = y. Linear Alg & Its Appl 96:109–129

    Article  MATH  Google Scholar 

  21. Lemke CE (1965) Bimatrix equilibrium points and mathematical programming. Managem Sci 11:681–689

    MathSciNet  Google Scholar 

  22. Lemke CE, Howson JT Jr (1964) Equilibrium points of bimatrix games. SIAM J Appl Math 12:413–423

    Article  MATH  Google Scholar 

  23. Mangasarian OL, Pang JS (1995) The extended linear complementarity problem. SIAM J Matrix Anal Appl 16:359–368

    Article  MathSciNet  MATH  Google Scholar 

  24. Mohan SR, Neogy SK (1997) Vertical block hidden Z-matrices and the generalized linear complementarity problem. SIAM J Matrix Anal Appl 18:181–190

    Article  MathSciNet  MATH  Google Scholar 

  25. Murty KG (1968) On the number of solutions to the complementarity problem and spanning properties of complementary cones. Univ. Calif., Berkeley, CA

    Google Scholar 

  26. Murty KG (1972) On the number of solutions to the complementarity problem and spanning properties of complementary cones. Linear Alg & Its Appl 5:65–108

    Article  MathSciNet  MATH  Google Scholar 

  27. Murty KG (1988) Linear complementarity: linear and nonlinear programming. Heldermann, Berlin

    MATH  Google Scholar 

  28. Pang JS (1995) Complementarity problems. In: Horst R, Pardalos PM (eds) Handbook Global Optim. Kluwer, Dordrecht, pp 271–338

    Google Scholar 

  29. Robinson SM (1979) Generalized equations and their solutions, Part I: Basic theory. Math Program Stud 10:128–141

    MATH  Google Scholar 

  30. Samelson H, Thrall RM, Wesler O (1958) A partition theorem for Euclidean n-space. Proc Amer Math Soc 9:805–807

    Article  MathSciNet  MATH  Google Scholar 

  31. De Schutter B, De Moor B (1996) The extended linear complementarity problem. Math Program 71:289–326

    MathSciNet  Google Scholar 

  32. Sznajder R, Gowda MS (1995) Generalizations of P 0- and P-properties; Extended vertical and horizontal LCPs. Linear Alg & Its Appl 223/224:695–715

    Article  MathSciNet  Google Scholar 

  33. Tütüncü RH, Todd MJ (1995) Reducing horizontal linear complementarity problems. Linear Alg & Its Appl 223/224:717–730

    Article  Google Scholar 

  34. Väliaho H (1996) P -matrices are just sufficient. Linear Alg & Its Appl 239:103–108

    MATH  Google Scholar 

  35. Website: www.cs.wisc.edu/cpnet

  36. Ye Y (1993) A fully polynomial-time approximation algorithm for computing a stationary point of the general linear complementarity problem. Math Oper Res 18:334–346

    MathSciNet  MATH  Google Scholar 

  37. Yoshise A (1996) Complementarity problems. In: Terlaky T (ed) Interior point methods of mathematical programming. Kluwer, Dordrecht, pp 297–367

    Google Scholar 

  38. Zhang Y (1994) On the convergence of a class of infeasible interior-point algorithm for the horizontal linear complementarity problem. SIAM J Optim 4:208–227

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2008 Springer-Verlag

About this entry

Cite this entry

Cottle, R.W. (2008). Linear Complementarity Problem . In: Floudas, C., Pardalos, P. (eds) Encyclopedia of Optimization. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-74759-0_333

Download citation

Publish with us

Policies and ethics