Description
Book SynopsisThis 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.
Table of Contents1. Introduction1.1 Motivation1.2 Historical antecedents1.3 Working framework and objectives
2. Preliminaries2.1 Functional analysis background2.2 Convexity and continuity2.3 Examples of convex functions2.4 Exercises2.5 Bibliographical notes
3. Fenchel-Moreau-Rockafellar theory3.1 Conjugation theory3.2 Fenchel-Moreau-Rockafellar theorem3.3 Dual representations of support functions
3.4 Minimax theory
3.5 Exercises
3.6 Bibliographical notes
4. Fundamental topics in convex analysis4.1 Subdifferential theory4.2 Convex duality4.3 Convexity in Banach spaces4.4 Subdifferential integration4.5 Exercises4.6 Bibliographical notes
5. Supremum of convex functions5.1 Conjugacy based approach5.2 Main subdifferential formulas 5.3 The role of continuity assumptions
5.4 Exercises
5.5 Bibliographical notes
6. The supremum in specific contexts6.1 The compact-continuous setting6.2 Compactification approach6.3 Main subdifferential formula revisited 6.4 Homogeneous formulas
6.5 Qualification conditions
6.6 Exercises
6.7 Bibliographical notes
7. Other subdifferential calculus rules7.1 Subdifferential of the sum7.2 Symmetric versus asymmetric conditions7.3 Supremum-sum subdifferential calculus 7.4 Exercises7.5 Bibliographical notes
8. Miscellaneous8.1 Convex systems and Farkas-type qualifications8.2 Optimality and duality in (semi)infinite convex optimization8.3 Convexification processes in optimization
8.4 Non-convex integration
8.5 Variational characterization of convexity
8.6 Chebychev sets and convexity
8.7 Exercises
8.8 Bibliographical notes
9. Exercises- Solutions9.1 Exercises of chapter 29.2 Exercises of chapter 39.3 Exercises of chapter 49.4 Exercises of chapter 59.5 Exercises of chapter 6
9.6 Exercises of chapter 7
9.7 Exercises of chapter 8
IndexGlossary of NotationsBibliography
