| This work is devoted to the study of certain normal compactness properties that play a crucial role in many aspects of variational analysis in infinite-dimensional spaces. Variational analysis has been recognized as an important area in nonlinear analysis mostly oriented on applications to constrained optimization and related problems. On the other hand, variational principles and methods of variational analysis are very useful for studying a broad spectrum of non-variational problems, in particular, various aspects of stability and sensitivity with respect to perturbations, metric regularity and openness properties, generalized differential calculus, etc. One of the principal parts of variational analysis is the generalized differentiation theory dealing with nonsmooth and generally nonconvex objects like sets, functions, and set-valued mappings (multifunctions), which appear naturally and frequently in many areas of analysis and optimization.;The development and applications of variational analysis in infinite-dimensional spaces require new concepts and tools that cannot be found in the finite-dimensional theory. In particular, one of the most crucial ingredients of the generalized differentiation and optimization theories, as well as metric regularity and stability issues in infinite-dimensions is the presence of new conditions that contain sufficient amounts of compactness for performing limiting procedures. Such conditions, which are automatic in finite dimensions, allow us to obtain efficient calculus rules of generalized differentiation and apply them to infinite-dimensional problems in optimization, stability, control, economics, etc.;Despite a crucial significance of such compactness properties for infinite-dimensional variational analysis and its applications, it has not been systematically studied yet how they behave under various operations performed on sets, functions, and set-valued mappings. In particular, it is important to find conditions ensuring the preservation of these properties under intersections of sets, sums and other compositions of set-valued mappings and functions.;In this study we generalize such compactness conditions associated with the concept of sequential normal compactness (SNC) formulated in Mordukhovich and Shao [25]. The latter notion for sets is closely related to the compactly epi-Lipschitzian property in the sense of Borwein and Strojwas [3], but it may be less restrictive in some situations. We will also demonstrate how these new conditions help us in the development of extremal principles and extended subdifferential calculus. Then we conduct a systematic study of sequential normal compactness. In fact, we develop a comprehensive SNC calculus for sets, set-valued mappings, and extended-real-valued functions that provides efficient conditions for the preservation of the SNC and related properties under various operations. To establish the main results, we employ a variational geometric approach based on the extremal principle that can be viewed as a variational counterpart of the classical separation principle in the case of nonconvex sets; see [22] for more details and references.;In the final chapter of this work, we establish both optimality and suboptimality necessary conditions for a broad class of nonsmooth optimization problems with the help of sequential normal compactness calculus developed in this study. |
