This book examines the most fundamental parts of convex analysis and its applications to optimization and location problems. Accessible techniques of variational analysis are employed to clarify and simplify some basic proofs in convex analysis and to build a theory of generalized differentiation for convex functions and sets in finite dimensions. The book serves as a bridge for the readers who have just started using convex analysis to reach deeper topics in the field. Detailed proofs are presented for most of the results in the book and also included are many figures and exercises for better understanding the material. Applications provided include both the classical topics of convex optimization and important problems of modern convex optimization, convex geometry, and facility location.
Author(s): Boris Mordukhovich, Nguyen Mau Nam
Series: Synthesis Lectures on Mathematics & Statistics
Edition: 2
Publisher: Springer
Year: 2023
Language: English
Pages: 312
City: Cham
Preface to the Second Edition
Preface to the First Edition
Acknowledgments
Contents
Glossary of Notation and Acronyms
Operations and Symbols
Spaces
Sets
Functions
Set-Valued Mappings
Acronyms
List of Figures
1 Convex Sets and Functions
1.1 Preliminaries
1.2 Convex Sets
1.3 Convex Functions
1.4 Continuity and Lipschitz Continuity of Convex Functions
1.5 The Distance Function
1.6 Convex Set-Valued Mappings and Optimal Value Functions
1.7 Exercises for This Chapter
2 Convex Separation and Some Consequences
2.1 Affine Hulls and Relative Interiors of Convex Sets
2.2 Strict Separation of Convex Sets
2.3 Separation and Proper Separation Theorems
2.4 Relative Interiors of Convex Graphs
2.5 Normal Cones to Convex Sets
2.6 Exercises for This Chapter
3 Convex Generalized Differentiation
3.1 Subgradients of Convex Functions
3.2 Subdifferential Calculus Rules
3.3 Mean Value Theorems
3.4 Coderivative Calculus
3.5 Subgradients of Optimal Value Functions
3.6 Generalized Differentiation for Polyhedral Convex Functions and Multifunctions
3.7 Exercises for This Chapter
4 Fenchel Conjugate and Further Topics in Subdifferentiation
4.1 Fenchel Conjugates
4.2 Support Functions and Support Intersection Rules
4.3 Conjugate Calculus Rules
4.4 Directional Derivatives
4.5 Subgradients of Supremum Functions
4.6 Exercises for This Chapter
5 Remarkable Consequences of Convexity
5.1 Characterizations of Differentiability
5.2 Carathéodory Theorem
5.3 Farkas Lemma
5.4 Radon Theorem and Helly Theorem
5.5 Extreme Points and Minkowski Theorem
5.6 Tangents to Convex Sets
5.7 Exercises for This Chapter
6 Minimal Time Functions and Related Issues
6.1 Horizon Cones
6.2 Minimal Time Functions and Their Properties
6.3 Minkowski Gauge Functions
6.4 Subgradients of Minimal Time Functions with Bounded Dynamics
6.5 Subgradients of Minimal Time Functions with Unbounded Dynamics
6.6 Exercises for This Chapter
7 Applications to Problems of Optimization and Equilibrium
7.1 Lower Semicontinuity and Existence of Minimizers
7.2 Optimality Conditions in Convex Minimization
7.3 Fenchel Duality in Optimization
7.4 Lagrangian Duality in Convex Optimization
7.5 Subgradient Methods in Nonsmooth Convex Optimization
7.6 Nash Equilibrium via Convex Analysis
7.7 Exercises for This Chapter
8 Applications to Location Problems
8.1 The Fermat-Torricelli Problem
8.2 The Weiszfeld Algorithm
8.3 Generalized Fermat-Torricelli Problems
8.4 Solving Planar Fermat-Torricelli Problems for Euclidean Balls
8.5 Generalized Sylvester Problems
8.6 Solving Planar Smallest Intersection Ball Problems
8.7 Exercises for This Chapter
A Solutions and Hints for Selected Exercises
References
Index
