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Solving a class of asymmetric variational inequalities by a new alternating direction method. (English) Zbl 0959.49009
Summary: The augmented Lagrangian method (also referred to as an alternating direction method) solves a class of Variational Inequalities (VI) via solving a series of sub-VI problems. The method is effective whenever the subproblems can be solved efficiently. However, the subproblem to be solved in each iteration of the augmented Lagrangian method itself is still a VI problem. It is essentially as difficult as the original one, the only difference is that the dimension of the subproblems is lower. In this paper, we propose a new alternating direction method for solving a class of monotone variational inequalities. In each iteration, the method solves a convex quadratic programming with simple constraints and a well-conditioned system of nonlinear equations. For such ‘easier’ subproblems, existing efficient numerical softwares are applicable. The effectiveness of the proposed method is demonstrated with an illustrative example.

##### MSC:
 49J40 Variational inequalities 65K10 Numerical optimization and variational techniques 90C20 Quadratic programming 90C25 Convex programming
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##### References:
 [1] Dafermos, S., Traffic equilibrium and variational inequalities, Transportation science, 14, 42-54, (1980) [2] Lemke, C.E., Bimatrix equilibrium points and mathematical programming, Management science, 11, 681-689, (1965) · Zbl 0139.13103 [3] Nagurney, A., Network economics, A variational inequality approach, (1993), Kluwer Academic Dordrecht · Zbl 0873.90015 [4] Nagurney, A.; Ramanujam, P., Transportation network policy modeling with goal targets and generalized penalty functions, Transportation science, 30, 3-13, (1996) · Zbl 0849.90055 [5] Taji, K.; Fukushima, M.; Ibaraki, T., A globally convergent Newton method for solving strongly monotone variational inequalities, Mathematical programming, 58, 369-383, (1993) · Zbl 0792.49007 [6] Harker, P.T.; Pang, J.S., Finite-dimensional variational inequality and nonlinear complementarity problems: A survey of theory, algorithms and applications, Mathematical programming, 48, 161-220, (1990) · Zbl 0734.90098 [7] Harker, P.T.; Xiao, B., Newton’s method for the nonlinear complementarity problem: A B-differentiable equation approach, Mathematical programming, 48, 339-357, (1990) · Zbl 0724.90071 [8] He, B.S., A projection and contraction method for a class of linear complementarity problem and its application in convex quadratic programming, Applied mathematics and optimization, 25, 247-262, (1992) · Zbl 0767.90086 [9] He, B.S., A new method for a class of linear variational inequalities, Mathematical programming, 66, 137-144, (1994) · Zbl 0813.49009 [10] He, B.S., A class of projection and contraction method for monotone variational inequalities, Appl. mathematics and optimization, 35, 69-76, (1997) · Zbl 0865.90119 [11] Isac, G., Complementarity problems, lecture notes in mathematics, 1528, (1992), Springer-Verlag Berlin [12] Nagurney, A.; Thore, S.; Pan, J., Spatial market policy modeling with goal targets, Operations research, 44, 393-406, (1996) · Zbl 0855.90030 [13] Noor, M.A., Some recent advances in variational inequalities, part I. basic concepts, New Zealand J. math., 26, 53-80, (1997) · Zbl 0886.49004 [14] Noor, M.A., Some recent advances in variational inequalities, part II. other concepts, New Zealand J. math., 26, 229-255, (1997) · Zbl 0889.49006 [15] Fukushima, M., Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems, Mathematical programming, 53, 99-110, (1992) · Zbl 0756.90081 [16] Noor, M.A., An implicit method for mixed variational inequalities, Appl. math. lett., 11, 4, 109-113, (1998) · Zbl 0941.49005 [17] Gabay, D., Applications of the method of multipliers to variational inequalities, (), 299-331 [18] Gabay, D.; Mercier, B., A dual algorithm for the solution of nonlinear variation problems via finite-element approximations, Computers math. applic., 2, 1, 17-40, (1976) · Zbl 0352.65034 [19] Glowinski, R., Numerical methods for nonlinear variational problems, (1984), Springer-Verlag New York · Zbl 0575.65123 [20] Glowinski, R.; Lions, J.L.; Tremolieres, R., Numerical analysis of variational inequalities, (1981), North-Holland Amsterdam · Zbl 0508.65029 [21] He, B.S.; Yang, H., Some convergence properties of a method of multipliers for linearly constrained monotone variational inequalities, Operation research letters, 23, 151-161, (1998) · Zbl 0963.49006 [22] Bazaraa, M.S.; Shetty, C.M., (), 438-447 [23] Dennis, J.E.; Schnabel, R.B., Numerical methods for unconstrained optimization and nonlinear equations, (1983), Prentice-Hall Englewood Cliffs, NJ · Zbl 0579.65058 [24] Fletcher, R., Practical methods of optimization, (1985), Wiley New York · Zbl 0905.65002 [25] Ortega, J.M.; Rheiboldt, W.C., Iterative solution of nonlinear equations in several variables, (1970), Academic Press New York · Zbl 0241.65046 [26] Rice, J.R., () [27] Pang, J.S., Error bounds in mathematical programming, Mathematical programming, 79, 299-332, (1997) · Zbl 0887.90165 [28] Yang, H., Multiple equilibrium behaviors and advanced traveler information systems with endogenous market penetration, Transportation research B, 32, 205-218, (1998) [29] He, B.S., Inexact implicit method for general monotone variational inequalities, Mathematical programming, 86, 199-217, (1999) · Zbl 0979.49006
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