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.
Author(s): Boris S. Mordukhovich, Nguyen Mau Nam
Series: Springer Series in Operations Research and Financial Engineering
Edition: 1
Publisher: Springer Nature Switzerland
Year: 2022
Language: English
Pages: 585
City: Cham, Switzerland
Tags: Topological Vector Spaces, Convex Sets, Convex Functions, Convex Analysis, Variational Analysis, Generalized Differentiation, Variational Methods of Nonlinear Analysis
Preface
Contents
1 FUNDAMENTALS
1.1 Topological Spaces
1.1.1 Definitions and Examples
1.1.2 Topological Interior and Closure of Sets
1.1.3 Continuity of Mappings
1.1.4 Bases for Topologies
1.1.5 Topologies Generated by Families of Mappings
1.1.6 Product Topology and Quotient Topology
1.1.7 Subspace Topology
1.1.8 Separation Axioms
1.1.9 Compactness
1.1.10 Connectedness and Disconnectedness
1.1.11 Net Convergence in Topological Spaces
1.2 Topological Vector Spaces
1.2.1 Basic Concepts in Topological Vector Spaces
1.2.2 Weak Topology and Weak* Topology
1.2.3 Quotient Spaces
1.3 Some Fundamental Theorems of Functional Analysis
1.3.1 Hahn-Banach Extension Theorem
1.3.2 Baire Category Theorem
1.3.3 Open Mapping Theorem
1.3.4 Closed Graph Theorem and Uniform Boundedness Principle
1.4 Exercises for Chapter 1
1.5 Commentaries to Chapter 1
2 BASIC THEORY OF CONVEXITY
2.1 Convexity of Sets
2.1.1 Basic Definitions and Elementary Properties
2.1.2 Operations on Convex Sets and Convex Hulls
2.2 Cores, Minkowski Functions, and Seminorms
2.2.1 Algebraic Interior and Linear Closure
2.2.2 Minkowski Gauges
2.2.3 Seminorms and Locally Convex Topologies
2.3 Convex Separation Theorems
2.3.1 Convex Separation in Vector Spaces
2.3.2 Convex Separation in Topological Vector Spaces
2.3.3 Convex Separation in Finite Dimensions
2.3.4 Extreme Points of Convex Sets
2.4 Convexity of Functions
2.4.1 Descriptions and Properties of Convex Functions
2.4.2 Convexity under Differentiability
2.4.3 Operations Preserving Convexity of Functions
2.4.4 Continuity of Convex Functions
2.4.5 Lower Semicontinuity and Convexity
2.5 Extended Relative Interiors in Infinite Dimensions
2.5.1 Intrinsic Relative and Quasi-Relative Interiors
2.5.2 Convex Separation via Extended Relative Interiors
2.5.3 Extended Relative Interiors of Graphs and Epigraphs
2.6 Exercises for Chapter 2
2.7 Commentaries to Chapter 2
3 CONVEX GENERALIZED DIFFERENTIATION
3.1 The Normal Cone and Set Extremality
3.1.1 Basic Definition and Normal Cone Properties
3.1.2 Set Extremality and Convex Extremal Principle
3.1.3 Normal Cone Intersection Rule in Topological Vector Spaces
3.1.4 Normal Cone Intersection Rule in Finite Dimensions
3.2 Coderivatives of Convex-Graph Mappings
3.2.1 Coderivative Definition and Elementary Properties
3.2.2 Coderivative Calculus in Topological Vector Spaces
3.2.3 Coderivative Calculus in Finite Dimensions
3.3 Subgradients of Convex Functions
3.3.1 Basic Definitions and Examples
3.3.2 Subdifferential Sum Rules
3.3.3 Subdifferential Chain Rules
3.3.4 Subdifferentiation of Maximum Functions
3.3.5 Distance Functions and Their Subgradients
3.4 Generalized Differentiation under Polyhedrality
3.4.1 Polyhedral Convex Separation
3.4.2 Polyhedral Normal Cone Intersection Rule
3.4.3 Polyhedral Calculus for Coderivatives and Subdifferentials
3.5 Exercises for Chapter 3
3.6 Commentaries to Chapter 3
4 ENHANCED CALCULUS AND FENCHEL DUALITY
4.1 Fenchel Conjugates
4.1.1 Definitions, Examples, and Basic Properties
4.1.2 Support Functions
4.1.3 Conjugate Calculus
4.2 Enhanced Calculus in Banach Spaces
4.2.1 Support Functions of Set Intersections
4.2.2 Refined Calculus Rules
4.3 Directional Derivatives
4.3.1 Definitions and Elementary Properties
4.3.2 Relationships with Subgradients
4.4 Subgradients of Supremum Functions
4.4.1 Supremum of Convex Functions
4.4.2 Subdifferential Formula for Supremum Functions
4.5 Subgradients and Conjugates of Marginal Functions
4.5.1 Computing Subgradients and Another Chain Rule
4.5.2 Conjugate Calculations for Marginal Functions
4.6 Fenchel Duality
4.6.1 Fenchel Duality for Convex Composite Problems
4.6.2 Duality Theorems via Generalized Relative Interiors
4.7 Exercises for Chapter 4
4.8 Commentaries to Chapter 4
5 VARIATIONAL TECHNIQUES AND FURTHER SUBGRADIENT STUDY
5.1 Variational Principles and Convex Geometry
5.1.1 Ekeland's Variational Principle and Related Results
5.1.2 Convex Extremal Principles in Banach Spaces
5.1.3 Density of ε-Subgradients and Some Consequences
5.2 Calculus Rules for ε-Subgradients
5.2.1 Exact Sum and Chain Rules for ε-Subgradients
5.2.2 Asymptotic ε-Subdifferential Calculus
5.3 Mean Value Theorems for Convex Functions
5.3.1 Mean Value Theorem for Continuous Functions
5.3.2 Approximate Mean Value Theorem
5.4 Maximal Monotonicity of Subgradient Mappings
5.5 Subdifferential Characterizations of Differentiability
5.5.1 Gâteaux and Fréchet Differentiability
5.5.2 Characterizations of Gâteaux Differentiability
5.5.3 Characterizations of Fréchet Differentiability
5.6 Generic Differentiability of Convex Functions
5.6.1 Generic Gâteaux Differentiability
5.6.2 Generic Fréchet Differentiability
5.7 Spectral and Singular Functions in Convex Analysis
5.7.1 Von Neumann Trace Inequality
5.7.2 Spectral and Symmetric Functions
5.7.3 Singular Functions and Their Subgradients
5.8 Exercises for Chapter 5
5.9 Commentaries to Chapter 5
6 MISCELLANEOUS TOPICS ON CONVEXITY
6.1 Strong Convexity and Strong Smoothness
6.1.1 Basic Definitions and Relationships
6.1.2 Strong Convexity/Strong Smoothness via Derivatives
6.2 Derivatives of Conjugates and Nesterov's Smoothing
6.2.1 Differentiability of Conjugate Compositions
6.2.2 Nesterov's Smoothing Techniques
6.3 Convex Sets and Functions at Infinity
6.3.1 Horizon Cones and Unboundedness
6.3.2 Perspective and Horizon Functions
6.4 Signed Distance Functions
6.4.1 Basic Definition and Elementary Properties
6.4.2 Lipschitz Continuity and Convexity
6.5 Minimal Time Functions
6.5.1 Minimal Time Functions with Constant Dynamics
6.5.2 Subgradients of Minimal Time Functions
6.5.3 Signed Minimal Time Functions
6.6 Convex Geometry in Finite Dimensions
6.6.1 Carathéodory Theorem on Convex Hulls
6.6.2 Geometric Version of Farkas Lemma
6.6.3 Radon and Helly Theorems on Set Intersections
6.7 Approximations of Sets and Geometric Duality
6.7.1 Full Duality between Tangent and Normal Cones
6.7.2 Tangents and Normals for Polyhedral Sets
6.8 Exercises for Chapter 6
6.9 Commentaries to Chapter 6
7 CONVEXIFIED LIPSCHITZIAN ANALYSIS
7.1 Generalized Directional Derivatives
7.1.1 Definitions and Relationships
7.1.2 Properties of Extended Directional Derivatives
7.2 Generalized Derivative and Subderivative Calculus
7.2.1 Calculus Rules for Subderivatives
7.2.2 Calculus of Generalized Directional Derivatives
7.3 Directionally Generated Subdifferentials
7.3.1 Basic Definitions and Some Properties
7.3.2 Calculus Rules for Generalized Gradients
7.3.3 Calculus of Contingent Subgradients
7.4 Mean Value Theorems and More Calculus
7.4.1 Mean Value Theorems for Lipschitzian Functions
7.4.2 Additional Calculus Rules for Generalized Gradients
7.5 Strict Differentiability and Generalized Gradients
7.5.1 Notions of Strict Differentiability
7.5.2 Single-Valuedness of Generalized Gradients
7.6 Generalized Gradients in Finite Dimensions
7.6.1 Rademacher Differentiability Theorem
7.6.2 Gradient Representation of Generalized Gradients
7.6.3 Generalized Gradients of Antiderivatives
7.7 Subgradient Analysis of Distance Functions
7.7.1 Regular and Limiting Subgradients of Lipschitzian Functions
7.7.2 Regular and Limiting Subgradients of Distance Functions
7.7.3 Subgradients of Convex Signed Distance Functions
7.8 Differences of Convex Functions
7.8.1 Continuous DC Functions
7.8.2 The Mixing Property of DC Functions
7.8.3 Locally DC Functions
7.8.4 Subgradients and Conjugates of DC Functions
7.9 Exercises for Chapter 7
7.10 Commentaries to Chapter 7
Glossary of Notation and Acronyms
Glossary of Notation and Acronyms
List of Figures
References
Subject Index
Index
