Convex Analysis and Beyond: Volume I: Basic Theory

This book provides a unified theory of convex functions, sets, and set-valued mappings in topological vector spaces, using variational analysis and set extremality. It covers local convex, Banach, and finite-dimensional settings, consolidating fundamental facts of generalized differential calculus to obtain novel results. It also explores nonconvex generalized differentiation and its applications, with an adaptable framework for researchers and students.

Format: Hardback
Length: 585 pages
Publication date: 25 April 2022
Publisher: Springer Nature Switzerland AG

This comprehensive book offers a unified theory of convex functions, sets, and set-valued mappings in topological vector spaces, encompassing locally convex, Banach, and finite-dimensional settings. The foundation of this theory lies in the powerful geometric approach of variational analysis, which revolves around set extremality and its characterizations in the presence of convexity. By employing this approach, the text consolidates the device of fundamental facts of generalized differential calculus to yield novel results for convex sets, functions, and set-valued mappings in both finite and infinite dimensions. Furthermore, it extends the scope of convex analysis beyond convexity, exploring nonconvex generalized differentiation and its applications.

The text is meticulously crafted with researchers and multiple levels of students in mind. It includes an abundance of exercises and figures tailored to graduate classes in mathematical sciences, making them accessible to advanced students in economics, engineering, and other applications. Additionally, chapters on convex analysis and optimization in finite-dimensional spaces cater to upper undergraduate students, while the work as a whole serves as an invaluable resource for mathematicians and applied scientists, particularly those with expertise in convex and variational analysis, optimization, and their applications.

The book's organization is designed to facilitate a seamless learning experience. It begins with an introduction to convex analysis and set-valued mappings, providing a solid foundation for the subsequent chapters. The subsequent chapters delve into the theory of convex functions, sets, and set-valued mappings, covering various topics such as convex optimization, duality theory, and applications in economics and engineering. Each chapter is accompanied by detailed explanations, examples, and exercises, allowing readers to grasp the concepts and apply them to real-world scenarios.

Moreover, the book incorporates numerous figures and diagrams to visually aid in understanding the concepts and theorems presented. These visual aids enhance the reader's comprehension and retention of the material, making the book an invaluable resource for both academic and professional purposes.

In conclusion, this book is a groundbreaking contribution to the field of convex and variational analysis. It provides a comprehensive and accessible introduction to the theory and applications of convex functions, sets, and set-valued mappings, making it an essential resource for students, researchers, and practitioners alike. By leveraging the power of geometric analysis and its applications, the book offers a new perspective on optimization and its practical implications, paving the way
way for further advancements in various fields of mathematics and applied sciences.

Weight: 1236g
Dimension: 235 x 155 (mm)
ISBN-13: 9783030947842
Edition number: 1st ed. 2022