Journal of Optimization Theory and Applications

, Volume 140, Issue 2, pp 233–238 | Cite as

EP Theorem for Dual Linear Complementarity Problems

Article

Abstract

The linear complementarity problem (LCP) belongs to the class of \(\mathbb{NP}\) -hard problems. Therefore, we cannot expect a polynomial time solution method for LCPs without requiring some special property of the matrix of the problem. We show that the dual LCP can be solved in polynomial time if the matrix is row sufficient; moreover, in this case, all feasible solutions are complementary. Furthermore, we present an existentially polytime (EP) theorem for the dual LCP with arbitrary matrix.

Keywords

Linear complementarity problem Dual LCP Row sufficient matrix *-matrix EP theorem 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Kojima, M., Megiddo, N., Noma, T., Yoshise, A.: A Unified Approach to Interior Point Algorithms for Linear Complementarity Problems. Lecture Notes in Computer Science, vol. 538. Springer, Berlin (1991) Google Scholar
  2. 2.
    Fukuda, F., Namiki, M., Tamura, A.: EP theorems and linear complementarity problems. Discrete Appl. Math. 84, 107–119 (1998) CrossRefMathSciNetGoogle Scholar
  3. 3.
    Fukuda, K., Terlaky, T.: Linear complementary and orientated matroids. J. Oper. Res. Soc. Jpn. 35, 45–61 (1992) MathSciNetGoogle Scholar
  4. 4.
    Cottle, R.W., Pang, J.-S., Venkateswaran, V.: Sufficient matrices and the linear complementarity problem. Linear Algebra Appl. 114/115, 231–249 (1989) CrossRefMathSciNetGoogle Scholar
  5. 5.
    Guu, S.-M., Cottle, R.W.: On a subclass of P 0. Linear Algebra Appl. 223/224, 325–335 (1995) CrossRefMathSciNetGoogle Scholar
  6. 6.
    Väliaho, H.: P *-matrices are just sufficient. Linear Algebra Appl. 239, 103–108 (1996) MathSciNetGoogle Scholar
  7. 7.
    Csizmadia, Zs., Illés, T.: New criss-cross type algorithms for linear complementarity problems with sufficient matrices. Optim. Methods Softw. 21, 247–266 (2006) CrossRefMathSciNetGoogle Scholar
  8. 8.
    Cameron, K., Edmonds, J.: Existentially polytime theorems. In: Polyhedral Combinatorics. DIMACS Series in Discrete Mathematics and Theoretical Computer Science, vol. 1, pp. 83–100. American Mathematical Society, Providence (1990) Google Scholar
  9. 9.
    Roos, C., Terlaky, T., Vial, J.-Ph.: Theory and Algorithms for Linear Optimization, An Interior Point Approach. Wiley-Interscience Series in Discrete Mathematics and Optimization. Wiley, New York (1997). 2nd edn. Interior Point Methods for Linear Optimization. Springer, New York (2006) Google Scholar
  10. 10.
    Illés, T., Nagy, M., Terlaky, T.: Interior point algorithms for general LCPs. J. Glob. Optim. (2008, to appear) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.Department of Management ScienceStrathclyde UniversityGlasgowUK
  2. 2.Department of Operation ResearchEötvös Lorànd University of ScienceBudapestHungary
  3. 3.Department of Computing and SoftwareSchool of Computational Engineering and Science, McMaster UniversityHamiltonCanada

Personalised recommendations