Abstract
In this paper, we propose a new operator splitting method for solving a class of variational inequality problems in which part of the underlying mappings are unknown. This class of problems arises frequently from engineering, economics and transportation equilibrium problems. At each iteration, by using the information observed from the system, the method solves a system of nonlinear equations, which is well-defined. Under mild assumptions, the global convergence of the method is proved, and its efficiency is demonstrated with numerical examples.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bertsekas D.P., Tsitsiklis J.N.: Parallel and Distributed Computation: Numerical Methods. Prentice-Hall, Englewood Cliffs (1989)
Button, K.J.: Transport Economics, 2nd edn. Edward Elgar, England
Douglas J., Rachford H.H.: On the numerical solution of the heat conduction problem in 2 and 3 space variables. Trans. Am. Math. Soc. 82, 421–439 (1956)
Eaves B.C.: Computing stationary points. Math. Program. Study 7, 1–14 (1978)
Eckstein J., Bertsekas D.P.: On the Douglas–Rachford splitting method and the proximal point algorithm for maximal monotone operators. Math. Program. 55, 293–318 (1992)
Facchinei F., Pang J.S.: Finite-dimensional variational inequalities and complementarity problems, Volumes I and II. Springer, Berlin (2003)
Fukushima M.: The primal Douglas–Rachford splitting algorithm for a class of monotone mappings with application to the traffic equilibrium problem. Math. Program. 72, 1–15 (1996)
Glowinski R.: Numerical methods for nonlinear variational problems. Springer, New York (1984)
Glowinski, R., Le Tallec, P.: Augmented Lagrangian and Operator-Splitting Method in Nonlinear Mechanics. SIAM Studies in Applied Mathematics, Philadelphia (1989)
Han D.R., Sun W.Y.: A new modified Goldstein–Levitin–Polyak projection method for variational inequality problems. Comput. Math. Appl. 47, 1817–1825 (2004)
Han D.R.: Inexact operator splitting methods with self-adaptive strategy for variational inequality problems. J. Optim. Theory Appl. 132, 227–243 (2007)
He, B.S.: Some predictor-corrector projection methods for monotone variational inequalities, Report 95-68, Faculty of Technical Mathematics and Informatics, University of technology, Delft (1995)
He B.S.: Inexact implicit methods for monotone general variational inequalities. Math. Program. 86, 199–217 (1999)
He B.S., Yang H., Meng Q., Han D.R.: Modified Goldstein–Levitin–Polyak projection method for asymmetric strongly monotone variational inequalities. J. Optim. Theory Appl. 112, 129–143 (2002)
He B.S., Liao L.Z., Wang S.L.: Self-adaptive operator splitting methods for monotone variational inequalities. Numerische Mathematik 94, 715–737 (2003)
Huang N.J.: A new method for a class of nonlinear variational inequalities with fuzzy mapping. Appl. Math. Lett. 10, 129–133 (1997)
Lions P.L., Mercier B.: Splitting algorithms for the sum of two nonlinear operators. SIAM J. Numer. Anal. 16, 964–979 (1979)
Liu L.N., Boyce D.: Variational inequality formulation of the system-optimal travel choice problem and efficient congestion tolls for a general transportation network with multiple time periods. Reg. Sci. Urban Econ. 32, 627–650 (2002)
Martinet B.: Regularization d’Inéquations Variationelles par Approximations Successives. Revue Fransaise d’Informatique et de Recherche Opérationelle 4, 154–159 (1970)
Nagurney A.: Network Economics: A Variational Inequality Approach. Kluwer, Dordrecht (1998)
Pang J.S.: Error bounds in mathematical programming. Math. Program. 79, 299–332 (1997)
Peaceman D.H., Rachford H.H.: The numerical solution of parabolic elliptic differential equations. J. Soc. Ind. Appl. Math. 3, 28–41 (1955)
Rockafellar R.T.: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 14, 877–898 (1976)
Samuelson P.A.: Spatial price equilibrium and linear programming. Am. Econ. Rev. 42, 283–303 (1952)
Takayama T., Judge G.G.: Spatial and Temporal Price and Allocation Models. North-Holland, Amsterdam (1971)
Varga R.S.: Matrix Iterative Analysis. Prentice-Hall, Englewood Cliffs (1966)
Xu, W., Yang, H., Han, D.R.: Sequential experimental approach for congestion pricing with multiple vehicle types and multiple time periods, Manuscript (2006)
Yang H., Meng Q., Lee D.H.: Trial-and-error implementation of marginal-cost pricing on networks in the absence of demand functions. Transp. Res. 38, 477–493 (2004)
Author information
Authors and Affiliations
Corresponding author
Additional information
The research of D. Han is supported by NSFC grant 10501024 and NSF of Jiangsu Province at Grant No. BK2006214.
Rights and permissions
About this article
Cite this article
Han, D., Xu, W. & Yang, H. An operator splitting method for variational inequalities with partially unknown mappings. Numer. Math. 111, 207–237 (2008). https://doi.org/10.1007/s00211-008-0181-7
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00211-008-0181-7