Mathematical Programming

, Volume 17, Issue 1, pp 335–344 | Cite as

Computational complexity of LCPs associated with positive definite symmetric matrices

  • Yahya Fathi


Murty in a recent paper has shown that the computational effort required to solve a linear complementarity problem (LCP), by either of the two well known complementary pivot methods is not bounded above by a polynomial in the size of the problem. In that paper, by constructing a class of LCPs—one of ordern forn ≥ 2—he has shown that to solve the problem of ordern, either of the two methods goes through 2 n pivot steps before termination.

However that paper leaves it as an open question to show whether or not the same property holds if the matrix,M, in the LCP is positive definite and symmetric. The class of LCPs in whichM is positive definite and symmetric is of particular interest because of the special structure of the problems, and also because they appear in many practical applications.

In this paper, we study the computational growth of each of the two methods to solve the LCP, (q, M), whenM is positive definite and symmetric and obtain similar results.

Key words

Linear Complementarity Problem Complementary Cones Complementary Pivot Methods Computational Complexity Exponential Growth 


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  1. [1]
    R.W. Cottle and G.B. Dantzig, “Complementary pivot theory of mathematical programming”,Linear Algebra and Its Applications 1 (1968) 103–125.Google Scholar
  2. [2]
    R.W. Cottle, “Monotone solutions of the parametric linear complementarity problem”,Mathematical Programming 3 (1972) 210–224.Google Scholar
  3. [3]
    V. Klee and G.L. Minty, “How good is the simplex algorithm”, in: O. Shisha, ed.,Inequalities III (Academic Press, New York, 1976).Google Scholar
  4. [4]
    M.M. Kostreva, “Direct algorithms for complementarity problems”, Dissertation, Rennselaer Polytechnic Institute (Troy, New York, 1976).Google Scholar
  5. [5]
    C.E. Lemke, “Bimatrix equilibrium points and mathematical programming”,Management Science 4 (1965) 681–689.Google Scholar
  6. [6]
    K.G. Murty,Linear and combinatorial programming (Wiley, New York, 1976).Google Scholar
  7. [7]
    K.G. Murty, “Computational complexity of complementary pivot methods”,Mathematical Programming Study 7 (1978) 61–73.Google Scholar
  8. [8]
    K.G. Murty, “On the number of solutions of the complementarity problems and spanning properties of complementary cones”,Linear Algebra and Its Applications 5 (1972) 65–108.Google Scholar
  9. [9]
    K.G. Murty, “On the parametric complementarity problem”,Summer conference notes (The University of Michigan, 1971).Google Scholar
  10. [10]
    K.G. Murty, “Note on a Bard-type scheme for solving the complementarity problem”,Opsearch 7 (1970) 263–268.Google Scholar
  11. [11]
    K.G. Murty, “On a characterization ofP-matrices”,SIAM Journal on Applied Mathematics 20 (3) (1971) 378–384.Google Scholar
  12. [12]
    R. Saigal, “On the class of complementary cones and Lemke's algorithm”,SIAM Journal on Applied Mathematics 23 (1972) 47–60.Google Scholar
  13. [13]
    A.C. Stickney and L.T. Watson, “Digraph models of Bard-type algorithms for the linear complementarity problem”,Mathematics of Operations Research 3 (1978) 322–333.Google Scholar

Copyright information

© North-Holland Publishing Company 1979

Authors and Affiliations

  • Yahya Fathi
    • 1
  1. 1.The University of MichiganAnn ArborUSA

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