Structure And Stability Of Restrictive Metric Regularity With Applications

It is well known that variational analysis emerges as a new branch ofmathematics in the past few decades following the development of mathematicalprogramming, optimal control, mathematical economics, game theory, approximationtheory, and calculus of variations, etc. As one of the most fundamental concepts inmodern variational analysis, metric regularity establishes its crucial role in boththeory and applications. This study involves the so-called restrictive metric regularity(RMR) that was first proposed by Mordukhovich and Wang in [32], where interestingapplications in generalized differentiation theory were established. Our main goal inthis study is to clarify the structure of RMR mappings, and then establish criteria ofRMR mappings and discuss their stabilities under perturbations. We also provideapplications of RMR mappings to generalized sequential normal compactness whichis another important concept in variational analysis.The study encompasses the following parts:Chapter1. Introduction and preliminaries involving RMR.Chapter2. Structure of RMR mappings. We establish necessary and sufficientconditions for the RMR property, i.e., the decompositions in subspaces, providediscussions of related theoretical issues and results. In case of finite dimensionalsituations, we also provide algorithms to determine the RMR property.Chapter3. Stability of RMR mappings under perturbations. Although metricregularity is usually stable under small perturbations, we will show in this chapter thatthe situation for RMR is much more involved. We will establish comprehensiveresults in both linear and nonlinear cases and provide detailed examples.Chapter4. Applications of RMR mappings to generalized sequential normalcompactness. The rule of inverse images and the chain rule are established.