Appl Math Optim 45:63–74 (2002)
2002 Springer-Verlag New York Inc.
A Modiﬁed Alternating Direction Method
for Variational Inequality Problems
Department of Mathematics, Nanjing University,
Nanjing 210093, People’s Republic of China
Communicated by J. Stoer
Abstract. The alternating direction method is an attractive method for solving
large-scale variational inequality problems whenever the subproblems can be solved
efﬁciently.However,thesubproblemsare still variationalinequalityproblems,which
are as structurally difﬁcult to solve as the original one. To overcome this disadvan-
tage, in this paper we propose a new alternating direction method for solving a class
of nonlinear monotone variational inequality problems. In each iteration the method
just makes an orthogonal projection to a simple set and some function evaluations.
We report some preliminary computational results to illustrate the efﬁciency of the
Key Words. Variational inequality problems, Alternating direction methods,
Monotone mappings, Global convergence.
AMS Classiﬁcation. 90C33, 90C30.
A classical variational inequality problem, denoted by VI( f, S), is to ﬁnd a vector x
(z − x
) ≥ 0, ∀z ∈ S, (1)
where S ⊂ R
is a nonempty closed convex subset of R
and f is a mapping from R
into itself. In trafﬁc equilibrium and network economics problems , , , S often
This research was supported by NSFC Grant 19971040.