On the Connectedness of Solution Sets of Parametrized Equations and of Solution Sets in Linear Complementarity Problems

  • M. Seetharama Gowda
  • G. S. R. Murthy
  • T. Parthasarathy
Part of the Applied Optimization book series (APOP, volume 50)


In this article, we prove, under certain conditions, the connectedness of sets of the form {x: f(x, y) = 0, y ∈ E} where f is a function with x varying over an open set in R n and the parameter y varying over a topological space. Based on this, we show that the partitioned matrix
$$M = \left[ \begin{gathered} A\,\,\,\,\,\,B \hfill \\ B\,\,\,\,\,\,D \hfill \\\end{gathered} \right]$$
is (LCP) connected (i.e., for all q, the solution set of LCP(q, M) is connected) when AP 0Q, C = 0, and D is connected. We also show that (a) any nonnegative P 0Q 0-matrix is connected and (b) any matrix M partitioned as above with C and D nonnegative, and AP 0Q is connected.


Linear complementarity problem solution sets connectedness weak univalence 


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  1. [1]
    C. Berge, Topological Spaces, Oliver and Boyd, First English Edition, Edinburgh and London, 1963.Google Scholar
  2. [2]
    A.K. Biswas and G.S.R. Murthy, “A chain condition for Q 0 -matrices,” in Game Theoretical Applications To Economics And Operations Research, Springer Science+Business Media Dordrecht, Netherlands, pp. 149–152, 1997.Google Scholar
  3. [3]
    M. Cao and M.C. Ferris, “Pc-matrices and the linear complementarity problem,” Linear Algebra and its Applications, vol. 246, pp. 231–249, 1996.MathSciNetCrossRefGoogle Scholar
  4. [4]
    R.W. Cottle, J.S. Pang and R.E. Stone, The Linear Complementarity Problem, Academic Press, New York, 1992.zbMATHGoogle Scholar
  5. [5]
    R.W. Cottle and R.E. Stone, “On the uniqueness of solutions to linear complementarity problems,” Mathematical Programming, vol. 27, pp. 191–213, 1983.MathSciNetzbMATHCrossRefGoogle Scholar
  6. [6]
    N. Eagambaram and S.R. Mohan, “On some classes of linear complementarity problems with matrices of order n and rank (n — 1),” Mathematics of Operations Research, vol. 15, pp. 243–257, 1990.MathSciNetzbMATHCrossRefGoogle Scholar
  7. [7]
    M.S. Gowda and R. Sznajder, “Weak univalence and connectedness of inverse images of continuous functions,” Mathematics of Oper-atations Research, vol. 24, pp. 255–261, 1999.MathSciNetzbMATHCrossRefGoogle Scholar
  8. [8]
    C. Jones and M.S. Gowda, “On the connectedness of solution sets of linear complementarity problems,” Linear Algebra and its Applications, vol. 272, pp. 33–44, 1998.MathSciNetzbMATHCrossRefGoogle Scholar
  9. [9]
    G.S.R. Murthy and T. Parthasarathy, “Some properties of fully semimonotone Q0-matrices,” SIAM Journal on Matrix Analysis and Applications, vol. 16(4), pp. 1268–1286, 1995.MathSciNetzbMATHCrossRefGoogle Scholar
  10. [10]
    G.S.R. Murthy, T. Parthasarathy and B. Sriparna, “On the solution sets of linear complementarity problems,” SIAM Journal on Matrix Analysis and Applications, to appear.Google Scholar
  11. [11]
    G. Ravindran and M.S. Gowda, “Regularization of Po-functions in box variational inequality problems,” Research Report, Department of Mathematics and Statistics, University of Maryland Baltimore County, Baltimore, Maryland 21250, October 9, 1997.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2001

Authors and Affiliations

  • M. Seetharama Gowda
    • 1
  • G. S. R. Murthy
    • 2
  • T. Parthasarathy
    • 3
  1. 1.Department of Mathematics and StatisticsUniversity of MarylandBaltimore County, BaltimoreUSA
  2. 2.Indian Statistical InstituteHabsiguda, HyderabadIndia
  3. 3.Indian Statistical InstituteNew DelhiIndia

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