Metric Regularity. Theory and Applications - a survey

Abstract: Metric regularity has emerged during last 2-3 decades as one of the central concepts of variational analysis. The roots of this concept go back to a circle of fundamental regularity ideas of classical analysis embodied in such results as the implicit function theorem, Banach open mapping theorem, theorems of Lyusternik and Graves, on the one hand, and the Sard theorem and the Thom-Smale transversality theory, on the other. The three principal themes that are in the focus of attention are: (a) regularity criteria (containing quantitative estimates for rates of regularity) including formal comparisons of their relative power and precision; (b) stability problems relating to the effect of perturbations of the mapping on its regularity properties, on the one hand, and to solutions of equations, inclusions etc. on the other; (c) role of metric regularity in analysis and optimization. All of them are studied at three levels of generality: the general theory for (set-valued) mappings between metric spaces is followed by a detailed study of Banach and finite dimensional theories. There is a number of new results, both theoretical and relating to applications, and some known results are supplied with new, usually simpler, proofs.