A new method for a class of linear variational inequalities. (English) Zbl 0813.49009

The author considers a class of linear variational inequalities of the form \[ u\in \Omega(\nu- u)^ T (Mu+ q)\geq 0,\quad\text{for all }\nu\in \Omega, \] where \(M\) is a positive semidefinite matrix, \(q\in \mathbb{R}^ n\) and \(\Omega\subset \mathbb{R}^ n\) is a closed convex set. A new iteration scheme for the numerical solution of this problem is given. Each iteration of this method consists only of a projection to a convex set and two matrix-vector multiplications.


49J40 Variational inequalities
90C33 Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming)
65K10 Numerical optimization and variational techniques
Full Text: DOI


[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. · Zbl 0332.49005 · doi:10.1090/S0002-9904-1975-13874-2
[2] P.G. Ciarlet,Introduction to Matrix Numerical Analysis and Optimization, Collection of Applied Mathematics for the Master’s Degree (Masson, Paris, 1982).
[3] R.W. Cottle and G.B. Dantzig, ”Complementary pivot theory of mathematical programming,”Linear Algebra and Its Applications 1 (1968) 103–125. · Zbl 0155.28403 · doi:10.1016/0024-3795(68)90052-9
[4] S. Dafermos, ”An iterative scheme for variational inequalities,”Mathematical Programming 26 (1983) 40–47. · Zbl 0506.65026 · doi:10.1007/BF02591891
[5] S.C. Fang, ”An iterative method for generalized complementarity problems,”IEEE Transactions on Automatic Control AC 25 (1980) 1225–1227. · Zbl 0483.49027 · doi:10.1109/TAC.1980.1102537
[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. · Zbl 0699.65054
[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. · Zbl 0767.90086 · doi:10.1007/BF01182323
[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. · Zbl 0676.90087 · doi:10.1007/BF01587074
[9] C.E. Lemke, ”Bimatrix equilibrium points and mathematical programming,”Management Science 11 (1965) 681–689. · Zbl 0139.13103 · doi:10.1287/mnsc.11.7.681
[10] C.E. Lemke and J.T. Howson, ”Equilibrium points of bimatrix games,”SIAM Review 12 (1964) 45–78. · Zbl 0128.14804
[11] D.G. Luenberger,Introduction to Linear and Nonlinear Programming (Addison-Wesley, Reading, MA, 1973). · Zbl 0297.90044
[12] O.L. Mangasarian, ”Solution of symmetric linear complementarity problems by iterative methods,”Journal of Optimization Theory and Applications 22 (1979) 465–485. · Zbl 0341.65049 · doi:10.1007/BF01268170
[13] S. Mizuno, ”A new polynomial time algorithm for a linear complementarity problems,”Mathematical Programming 56 (1992) 31–43. · Zbl 0769.90077 · doi:10.1007/BF01580891
[14] J.S. Pang and D. Chan, ”Iterative methods for variational and complementarity problems,”Mathematical Programming 24 (1982) 284–313. · Zbl 0499.90074 · doi:10.1007/BF01585112
[15] J.S. Pang, ”Variational inequality problems over productsets: applications and iterative methods,”Mathematical Programming 31 (1985) 206–219. · Zbl 0578.49006 · doi:10.1007/BF02591749
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.