A Unified Theory for Metric Regularity of Multifunctions

We survey a large bunch of results on metric regularity of multifunctions that appeared during the last 25 years. The tools used for this survey rely on a new variational method introduced by D. Azé, J.-N. Corvellec and R. E. Lucchetti [Nonlinear Analysis 49 (2002) 643--670] and further developped by D. Azé and J.-N. Corvellec [ESAIM Control Optim. Calc. Var. 10 (2004) 409--425] and independently by A. D. Ioffe [in: "Approximation, Optimization and Mathematical Economics" (Guadeloupe, 1999), 165--176, M. Lassonde (ed.), Physica-Verlag, Heidelberg, 2001; and Russian Math. Surveys 55 (2000) 501--558] which provides a characterization of metric regularity. It allows us to give a unified point of view both for primal results (based on some notion of tangent cones) and dual ones (based on some notion of normal cones). For most known results on metric regularity, a simple proof is given along with some slight improvements for some of them