Appl Math Optim 43:147–168 (2001)
2001 Springer-Verlag New York Inc.
Convergence Properties of Projection and Contraction Methods
for Variational Inequality Problems
and J. Zhang
Department of Applied Mathematics, Northern Jiaotong University,
Beijing 100044, People’s Republic of China
Institute of Operations Research, Qufu Normal University,
Qufu 273165, People’s Republic of China
Department of Mathematics, City University of Hong Kong,
Kowloon, Hong Kong
Communicated by J. Stoer
Abstract. In this paper we develop the convergence theory of a general class of
projection and contraction algorithms (PC method), where an extended stepsize rule
is used, for solving variational inequality (VI) problems. It is shownthat, by deﬁning
a scaled projection residue, the PC method forces the sequence of the residues to
zero. It is also shown that, by deﬁning a projected function, the PC method forces
the sequence of projected functions to zero. A consequence of this result is that if
the PC method converges to a nondegenerate solution of the VI problem, then after
a ﬁnite number of iterations, the optimal face is identiﬁed. Finally, we study local
convergence behavior of the extragradient algorithm for solving the KKT system of
the inequality constrained VI problem.
Key Words. Variational inequality, Projection and contraction method, Predictor-
corrector stepsize, Convergence property.
AMS Classiﬁcation. 90C30, 90C33, 65K05.
This research was supported by the National Natural Science Foundation of China (19971002,
19871049), and CityU Strategic Research Grant.