这本书介绍了拓扑向量空间中凸函数、集和集值映射的统一理论，以及局部凸、Banach和有限维设置的规范。这些发展和论述基于变分分析的强大几何方法，该方法依赖于集极值及其在凸性存在下的特征和规格。利用这种方法，本文整合了广义微分学的基本事实，以获得有限维和无限维凸集、函数和集值映射的新结果。它还探索了凸性之外的主题，使用凸分析的基本机制来开发非凸广义微分及其应用。本文采用了一个适应性强的框架，由研究人员和多个层次的学生共同设计。它包括许多适合数学科学研究生班的练习和数字，也适用于经济学、工程学和其他应用领域的高级学生。此外，它还包括关于有限维空间中的凸分析和优化的章节，这些章节将对高年级本科生有用，而这项工作作为一个整体，为数学家和应用科学家，尤其是凸分析和变分分析、优化及其应用方面的专家提供了丰富的资源。
Convex Analysis and Beyond: Volume I: Basic Theory
This book presents a unified theory of convex functions, sets, and set-valued mappings in topological vector spaces with its specifications to locally convex, Banach and finite-dimensional settings. These developments and expositions are based on the powerful geometric approach of variational analysis, which resides on set extremality with its characterizations and specifications in the presence of convexity. Using this approach, the text consolidates the device of fundamental facts of generalized differential calculus to obtain novel results for convex sets, functions, and set-valued mappings in finite and infinite dimensions. It also explores topics beyond convexity using the fundamental machinery of convex analysis to develop nonconvex generalized differentiation and its applications. The text utilizes an adaptable framework designed with researchers as well as multiple levels of students in mind. It includes many exercises and figures suited to graduate classes in mathematical sciences that are also accessible to advanced students in economics, engineering, and other applications. In addition, it includes chapters on convex analysis and optimization in finite-dimensional spaces that will be useful to upper undergraduate students, whereas the work as a whole provides an ample resource to mathematicians and applied scientists, particularly experts in convex and variational analysis, optimization, and their applications.
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