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A new method for a class of linear variational inequalities

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Abstract

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.

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References

  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. 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. 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. S. Dafermos, “An iterative scheme for variational inequalities,”Mathematical Programming 26 (1983) 40–47.

    Google Scholar 

  5. S.C. Fang, “An iterative method for generalized complementarity problems,”IEEE Transactions on Automatic Control AC 25 (1980) 1225–1227.

    Google Scholar 

  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. 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. 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. C.E. Lemke, “Bimatrix equilibrium points and mathematical programming,”Management Science 11 (1965) 681–689.

    Google Scholar 

  10. C.E. Lemke and J.T. Howson, “Equilibrium points of bimatrix games,”SIAM Review 12 (1964) 45–78.

    Google Scholar 

  11. D.G. Luenberger,Introduction to Linear and Nonlinear Programming (Addison-Wesley, Reading, MA, 1973).

    Google Scholar 

  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. S. Mizuno, “A new polynomial time algorithm for a linear complementarity problems,”Mathematical Programming 56 (1992) 31–43.

    Google Scholar 

  14. J.S. Pang and D. Chan, “Iterative methods for variational and complementarity problems,”Mathematical Programming 24 (1982) 284–313.

    Google Scholar 

  15. J.S. Pang, “Variational inequality problems over productsets: applications and iterative methods,”Mathematical Programming 31 (1985) 206–219.

    Google Scholar 

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This work is supported by the National Natural Science Foundation of the P.R. China and NSF of Jiangsu.

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He, B. A new method for a class of linear variational inequalities. Mathematical Programming 66, 137–144 (1994). https://doi.org/10.1007/BF01581141

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  • DOI: https://doi.org/10.1007/BF01581141

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