Approximate Solutions of Variational Inequalities and the Ekeland Principle

Let K be a closed convex subset of a real Banach space X, and let F be a map from X to its dual X*. We study the so-called variational inequality problem: given y∈X*,, does there exist x0∈K such that (in standard notation) F(x0)−y,x−x0≥0 for all x∈K? After a short exposition of work in this area, we establish conditions on F sufficient to ensure a positive answer to the question of whether F is a gradient operator. A novel feature of the proof is the key role played by the well-known Ekeland variational principle.