Abstract
The alternating direction method is an attractive method for a class of variational inequality problems if the subproblems can be solved efficiently. However, solving the subproblems exactly is expensive even when the subproblem is strongly monotone or linear. To overcome this disadvantage, this paper develops a new alternating direction method for cocoercive nonlinear variational inequality problems. To illustrate the performance of this approach, we implement it for traffic assignment problems with fixed demand and for large-scale spatial price equilibrium problems.
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BERTSEKAS, D. P., and GAFNI, E. M., Projection Method for Variational Inequalities with Applications to the Traffic Assignment Problem, Mathematical Programming Study, Vol. 17, pp. 139–159, 1987.
DAFERMOS, S., Traffic Equilibrium and Variational Inequalities, Transportation Science, Vol. 14, pp. 42–54, 1980.
HARKER, P. T., and PANG, J. S., Finite-Dimensional Variational Inequality and Nonlinear Complementarity Problems: A Surûey of Theory, Algorithms, and Applications, Mathematical Programming, Vol. 48, pp. 161–220, 1990. JOTA: VOL. 112, NO. 3, MARCH 2002 560
NAGURNEY, A., and RAMANUJAM, P., Transportation Network Policy Modeling with Goal Targets and Generalized Penalty Functions, Transportation Science, Vol. 30, pp. 3–13, 1996.
HE, B. S., A Projection and Contraction Method for a Class of Linear Complementary Problems and Its Applications to Conûex Quadratic Programming, Applied Mathematics and Optimization, Vol. 25, pp. 247–262, 1992.
HE, B. S., Further Deûelopments in an Iteratiûe Projection and Contraction Method for Linear Programming, Journal of Computational Mathematics, Vol. 11, pp. 350–364, 1993.
TAJI, K., FUKUSHIMA, M., and IBARAKI, T., A Globally Conûergent Newton Method for Solûing Strongly Monotone Variational Inequalities, Mathematical Programming, Vol. 58, pp. 369–383, 1993.
GABAY, D., Applications of the Method of Multipliers to Variational Inequalities, Augmented Lagrangian Methods: Applications to the Numerical Solution of Boundary-Value Problems, Edited by M. Fortin and R. Glowinski, North-Holland, Amsterdam, Holland, pp. 299–331, 1983.
GABAY, D., and MERCIER, B., A Dual Algorithm for the Solution of Nonlinear Variational Problems ûia Finite-Element Approximations, Computer and Mathematics with Applications, Vol. 2, pp. 17–40, 1976.
HE, B. S., and ZHOU, J., A Modified Alternating Direction Method for Conûex Minimization Problems, Applied Mathematics Letters, Vol. 13, pp. 123–130, 2000.
ZHU, D. L., and MARCOTTE, P., Cocoerciûity and Its Role in the Conûergence of Iteratiûe Schemes for Solûing Variational Inequalities, SIAM Journal on Optimization, Vol. 6, pp. 714–726, 1996.
COHEN, G., Auxiliary Problem Principle Extended to Variational Inequalities, Journal of Optimization Theory and Applications, Vol. 59, pp. 325–333, 1988.
EAVES, B. C., On the Basic Theorem of Complementarity, Mathematical Programming, Vol. 1, pp. 68–75, 1971.
NGUYEN, S., and DUPUIS, C., An Efficient Method for Computing Traffic Equilibria in Networks with Asymmetric Transportation Costs, Transportation Science, Vol. 18, pp. 185–202, 1984.
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Han, D., Lo, H. New Alternating Direction Method for a Class of Nonlinear Variational Inequality Problems. Journal of Optimization Theory and Applications 112, 549–560 (2002). https://doi.org/10.1023/A:1017964015910
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DOI: https://doi.org/10.1023/A:1017964015910