Abstract
The variational inequality problem provides a unified framework for a large
family of problems such as complementarity problems, minimization problems,
Nash games, vector minimization problems, to name a few. Although a large
number of solution algorithms have been developed for this problem, there is
still a wide scope for improvement and a need for extensive additional
research in this realm. In particular, efficient and convergent algorithms
for solving such problems are still being sought. With above motivations, we
introduce two algorithms for solving the variational inequality problem with
pseudomonotone operators in Banach spaces which consist of two principal
steps and admit an interesting geometric interpretation. In the first
algorithm, at a certain iterate, an auxiliary point is first computed
through a line search. Using this auxiliary point and the previous iterate,
we construct an appropriate hyperplane separating the solution set of the
given problem from the current iterate. Considering this information, the
next iterate is then defined as the Bregman projection of a certain point
onto the feasible set of the problem. Assuming standard hypotheses, we show
that the generated sequence by the algorithm weakly and globally converges
to some solution of the problem. In the second algorithm, we study the
issue of upgrading from weak to strong convergence, for which the Bregman
projection step in the above algorithm is replaced by the Bregman projection
of the initial point onto the intersection of the feasible set of the
problem and two proper halfspaces, built at the current iterate. Besides the
above weak convergence result versus the strong one, the latter algorithm
has a distinctive feature meaning that the generated sequence strongly and
globally converges to the closest solution of the problem to the initial
point which is of interest even in finite dimensional spaces.
Information:
Date:  Tuesday, January 5, 2010, 15:0017:00  Place:  Niavaran Bldg., Niavaran Square, Tehran, Iran 
