A new condition characterizing solutions of variational inequality problems
B. F. Svaiter | Gárciga Otero, Rolando
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.