Mathematical Programming

, Volume 66, Issue 1–3, pp 137–144 | Cite as

A new method for a class of linear variational inequalities

  • Bingsheng He


In this paper we introduce a new iterative scheme for the numerical solution of a class of linear variational inequalities. Each iteration of the method consists essentially only of a projection to a closed convex set and two matrix-vector multiplications. Both the method and the convergence proof are very simple.


Linear variational inequality Linear complementarity problem Projection 


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  1. [1]
    R.E. Bruck, “An iterative solution of a variational inequality for certain monotone operators in Hilbert space,”Bulletin of the American Mathematical Society 81 (1975) 890–892.Google Scholar
  2. [2]
    P.G. Ciarlet,Introduction to Matrix Numerical Analysis and Optimization, Collection of Applied Mathematics for the Master's Degree (Masson, Paris, 1982).Google Scholar
  3. [3]
    R.W. Cottle and G.B. Dantzig, “Complementary pivot theory of mathematical programming,”Linear Algebra and Its Applications 1 (1968) 103–125.Google Scholar
  4. [4]
    S. Dafermos, “An iterative scheme for variational inequalities,”Mathematical Programming 26 (1983) 40–47.Google Scholar
  5. [5]
    S.C. Fang, “An iterative method for generalized complementarity problems,”IEEE Transactions on Automatic Control AC 25 (1980) 1225–1227.Google Scholar
  6. [6]
    P.T. Harker and J.S. Pang, “A damped-Newton method for the linear complementarity problem,”Lectures in Applied Mathematics 26 (1990) 265–284.Google Scholar
  7. [7]
    B.S. He “A projection and contraction method for a class of linear complementarity problems and its application in convex quadratic programming,”Applied Mathematics and Optimization 25 (1992) 247–262.Google Scholar
  8. [8]
    M. Kojima, S. Mizuno and A. Yoshise, “A polynomial-time algorithm for a class of linear complementarity problems,”Mathematical Programming 44 (1989) 1–26.Google Scholar
  9. [9]
    C.E. Lemke, “Bimatrix equilibrium points and mathematical programming,”Management Science 11 (1965) 681–689.Google Scholar
  10. [10]
    C.E. Lemke and J.T. Howson, “Equilibrium points of bimatrix games,”SIAM Review 12 (1964) 45–78.Google Scholar
  11. [11]
    D.G. Luenberger,Introduction to Linear and Nonlinear Programming (Addison-Wesley, Reading, MA, 1973).Google Scholar
  12. [12]
    O.L. Mangasarian, “Solution of symmetric linear complementarity problems by iterative methods,”Journal of Optimization Theory and Applications 22 (1979) 465–485.Google Scholar
  13. [13]
    S. Mizuno, “A new polynomial time algorithm for a linear complementarity problems,”Mathematical Programming 56 (1992) 31–43.Google Scholar
  14. [14]
    J.S. Pang and D. Chan, “Iterative methods for variational and complementarity problems,”Mathematical Programming 24 (1982) 284–313.Google Scholar
  15. [15]
    J.S. Pang, “Variational inequality problems over productsets: applications and iterative methods,”Mathematical Programming 31 (1985) 206–219.Google Scholar

Copyright information

© The Mathematical Programming Society, Inc. 1994

Authors and Affiliations

  • Bingsheng He
    • 1
  1. 1.Department of MathematicsUniversity of NanjingNanjingChina

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