In this thesis, we construct a general duality scheme for monotone variational inequality problems. This scheme is analogous to the classical duality scheme in convex programming in the sense that the duality is obtained by adding perturbation variables. In order to reach this goal, we have before deepened some properties and characterizations of monotone and maximal monotone multi-valued maps (subsets) on a global and a local point of view. In particular, we give an algorithm for constructing a maximal monotone extension of an arbitrary monotone map. We have specifically studied monotone affine subspaces. In this particular case, the construction of a maximal monotone extension can be processed within a finite number of steps. Finally, applications of our duality scheme to some classes of variational inequality problems are discussed.
