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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

$u\in {\Omega }{\left(\nu -u\right)}^{T}\left(Mu+q\right)\ge 0,\phantom{\rule{1.em}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}\text{all}\phantom{\rule{4.pt}{0ex}}\nu \in {\Omega },$

where $M$ is a positive semidefinite matrix, $q\in {ℝ}^{n}$ and ${\Omega }\subset {ℝ}^{n}$ 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.

##### MSC:
 49J40 Variational methods including variational inequalities 90C33 Complementarity and equilibrium problems; variational inequalities (finite dimensions) 65K10 Optimization techniques (numerical methods)
##### Keywords:
linear variational inequalities; iteration scheme
##### 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. · 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. [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. [11] D.G. Luenberger,Introduction to Linear and Nonlinear Programming (Addison-Wesley, Reading, MA, 1973). [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