This book aims at an innovative approach within the framework of convex analysis and optimization, based on an in-depth study of the behavior and properties of the supremum of families of convex functions. It presents an original and systematic treatment of convex analysis, covering standard results and improved calculus rules in subdifferential analysis. The tools supplied in the text allow a direct approach to the mathematical foundations of convex optimization, in particular to optimality and duality theory. Other applications in the book concern convexification processes in optimization, non-convex integration of the Fenchel subdifferential, variational characterizations of convexity, and the study of Chebychev sets. At the same time, the underlying geometrical meaning of all the involved concepts and operations is highlighted and duly emphasized. A notable feature of the book is its unifying methodology, as well as the novelty of providing an alternative or complementary view to the traditional one in which the discipline is presented to students and researchers. This textbook can be used for courses on optimization, convex and variational analysis, addressed to graduate and post-graduate students of mathematics, and also students of economics and engineering. It is also oriented to provide specific background for courses on optimal control, data science, operations research, economics (game theory), etc. The book represents a challenging and motivating development for those experts in functional analysis, convex geometry, and any kind of researchers who may be interested in applications of their work.
Author(s): Rafael Correa, Abderrahim Hantoute, Marco A. López
Series: Springer Series in Operations Research and Financial Engineering
Publisher: Springer
Year: 2023
Language: English
Pages: 450
City: Cham
Preface
Contents
1 Introduction
1.1 Motivation
1.2 Historical antecedents
1.3 Working framework and objectives
2 Preliminaries
2.1 Functional analysis background
2.2 Convexity and continuity
2.3 Examples of convex functions
2.4 Exercises
2.5 Bibliographical notes
3 Fenchel–Moreau–Rockafellar theory
3.1 Conjugation theory
3.2 Fenchel–Moreau–Rockafellar theorem
3.3 Dual representations of support …
3.4 Minimax theory
3.5 Exercises
3.6 Bibliographical notes
4 Fundamental topics in convex analysis
4.1 Subdifferential theory
4.2 Convex duality
4.3 Convexity in Banach spaces
4.4 Subdifferential integration
4.5 Exercises
4.6 Bibliographical notes
5 Supremum of convex functions
5.1 Conjugacy-based approach
5.2 Main subdifferential formulas
5.3 The role of continuity assumptions
5.4 Exercises
5.5 Bibliographical notes
6 The supremum in specific contexts
6.1 The compact-continuous setting
6.2 Compactification approach
6.3 Main subdifferential formula …
6.4 Homogeneous formulas
6.5 Qualification conditions
6.6 Exercises
6.7 Bibliographical notes
7 Other subdifferential calculus rules
7.1 Subdifferential of the sum
7.2 Symmetric versus asymmetric …
7.3 Supremum-sum subdifferential …
7.4 Exercises
7.5 Bibliographical notes
8 Miscellaneous
8.1 Convex systems and Farkas-type …
8.2 Optimality and duality in …
8.3 Convexification processes in …
8.4 Non-convex integration
8.5 Variational characterization of …
8.6 Chebychev sets and convexity
8.7 Exercises
8.8 Bibliographical notes
9 Exercises - Solutions
9.1 Exercises of chapter 2摥映數爠eflinkchone22
9.2 Exercises of chapter 3摥映數爠eflinkch2a33
9.3 Exercises of chapter 4摥映數爠eflinkch2b44
9.4 Exercises of chapter 5摥映數爠eflinkch355
9.5 Exercises of chapter 6摥映數爠eflinkch4a66
9.6 Exercises of chapter 7摥映數爠eflinkch477
9.7 Exercises of chapter 8摥映數爠eflinkch688
Appendix Glossary of notations
Appendix Bibliography
Index
