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Some convergence properties of a method of multipliers for linearly constrained monotone variational inequalities. (English) Zbl 0963.49006
Summary: Variational inequalities have important application in mathematical programming. The alternative direction methods are suitable and often used in the literature in solving large-scale, linearly constrained variational inequalities arising in transportation research. In this paper, we present a few inequalities associated with the alternative direction method of multipliers given by {\it D. Gabay} and {\it B. Mercier} [Comput. Math. Appl. 2, 17-40 (1976; Zbl 0352.65034)]. The inequalities are helpful in understanding the algorithm.

49J40Variational methods including variational inequalities
90C33Complementarity and equilibrium problems; variational inequalities (finite dimensions)
Full Text: DOI
[1] A. Nagurney, Network Economics, A Variational Inequality Approach, Kluwer Academic Publishers, Dordrecht, 1993. · Zbl 0873.90015
[2] Nagurney, A.; Ramanujam, P.: Transportation network policy modeling with goal targets and generalized penalty functions. Transportation sci. 30, 3-13 (1996) · Zbl 0849.90055
[3] Nagurney, A.; Thore, S.; Pan, J.: Spatial market policy modeling with goal targets. Oper. res. 44, 393-406 (1996) · Zbl 0855.90030
[4] Gabay, D.; Mercier, B.: A dual algorithm for the solution of nonlinear variational problems via finite-element approximations. Computer and mathematics with applications 2, 17-40 (1976) · Zbl 0352.65034
[5] Lawphongpanich, S.; Hearn, D. W.: Benders decomposition for variational inequalities. Math. programming 48, 231-247 (1990) · Zbl 0722.90024
[6] D. Gabay, Applications of the method of multipliers to variational inequalities, in: M. Fortin, R. Glowinski (Eds.), Augmented Lagrange Methods: Applications to the Solution of Boundary-valued Problems, North-Holland, Amsterdam, Netherlands, 1983, pp. 299--331.
[7] M. Fortin, R. Glowinski (Eds.), Augmented Lagrangian Methods: Applications to the solution of Boundary-Valued Problems, North-Holland, Amsterdam, 1983.
[8] R. Glowinski, Numerical Methods for Nonlinear Variational Problems, Springer, New York, 1984. · Zbl 0536.65054
[9] Lions, P. L.; Mercier, B.: Splitting algorithms for the sum of two nonlinear operators. SIAM J. Numer. anal. 16, 964-979 (1979) · Zbl 0426.65050
[10] P. Tseng, Applications of splitting algorithm to decomposition in convex programming and variational inequalities, Technical Report, Laboratory for Information and Decision Systems, M.I.T., November 1988.
[11] Fukushima, M.: A relaxed projection method for variational inequalities. Math. programming 35, 58-70 (1986) · Zbl 0598.49024
[12] Harker, P. T.; Pang, J. S.: Finite-dimensional variational inequality and nonlinear complementarity problemsa survey of theory, algorithms and applications. Math. programming 48, 161-220 (1990) · Zbl 0734.90098
[13] Harker, P. T.; Xiao, B.: Newton’s method for the nonlinear complementarity problema B-differentiable equation approach. Math. programming 48, 339-357 (1990) · Zbl 0724.90071
[14] He, B. S.: A projection and contraction method for a class of linear complementarity problem and its application in convex quadratic programming. Appl. math. Optim. 25, 247-262 (1992) · Zbl 0767.90086
[15] He, B. S.: A new method for a class of linear variational inequalities. Math. programming 66, 137-144 (1994) · Zbl 0813.49009
[16] He, B. S.: Solving a class of linear projection equations. Numer. math. 68, 71-80 (1994) · Zbl 0822.65040
[17] He, B. S.: A class of new methods for monotone variational inequalities. Appl. math. Optim. 35, 69-76 (1997) · Zbl 0865.90119
[18] D. Kinderlehhrer, G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, Academic Press, New York, 1980.
[19] Pang, J. S.; Chan, D.: Iterative methods for variational and complementarity problems. Math. programming 24, 284-313 (1982) · Zbl 0499.90074
[20] Sun, D.: A projection and contraction method for nonlinear complementarity problem and its extensions. Math. numer. Sinica 16, 183-194 (1994) · Zbl 0900.65188
[21] Sun, D.: A class of iterative methods for solving nonlinear projection equations. J. optim. Theory appl. 91, 123-140 (1996) · Zbl 0871.90091
[22] Xiao, B.; Harker, P. T.: A nonsmooth Newton method for variational inequalities, itheory. Math. programming 65, 151-194 (1994) · Zbl 0812.65048
[23] Xiao, B.; Harker, P. T.: A nonsmooth Newton method for variational inequalities, iinumerical results. Math. programming 65, 195-216 (1994) · Zbl 0812.65049
[24] Eaves, B. C.: On the basic theorem of complementarity. Math. programming 1, 68-75 (1971) · Zbl 0227.90044