Preprint A387/2005
A new condition characterizing solutions of variational inequality problems

B. F. Svaiter | Gárciga Otero, Rolando

**Keywords: **
optimality conditions | variational inequalities | approximated solutions

This paper is devoted to the study of a new alternative necessary condition in variational inequality problems: AGP.
A feasible point satisfies such
condition if it is the limit of a sequence of
approximated solutions
of approximations of the variational problem. This condition
comes from optimization where the error
in the approximated solution is measured by the
projected gradient onto the approximated feasible set,
which is obtained from a linearization of the restrictions
with slacks to make the current point feasible. Thus, coined as
Approximated Gradient Projection (AGP) condition.
We state the AGP condition for variational inequality problems and show that
it is necessary for a point being a solution even without constraint
qualifications (e.g., Abadie's). Moreover, the AGP condition is sufficient in
continuous variational inequalities. Sufficiency also holds for variational
inequalities involving maximal monotone operators provided boundedness of the
vectors in the image of the operator (playing the roll of the gradients). Since AGP
is a condition verified by a sequence, it is particularly interesting
for iterative methods.