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E-raamat: Convex Analysis and Beyond: Volume I: Basic Theory

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Convex Analysis and Beyond: Volume I: Basic TheorySuurem pilt
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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.

Arvustused

Each chapter ends with an exercise section . While primarily addressed to researchers, the book can be used for graduate courses in optimization, by undergraduate and graduate students for theses and projects as well as by researchers and practitioners from other fields where tools from convex analysis, variational analysis and optimization play a role. All in one, the reviewer warmly recommends this book to anyone interested. (Sorin-Mihai Grad, zbMATH 1506.90001, 2023) This outstanding book will certainly be useful to anyone interested to learn convex analysis, in particular to graduate students and researchers in the field. Most parts of it can also serve as the basis of advanced courses on a variety of topics. In view of the excellence of this first volume, one can expect the best of the announced second one, which will deal with applications of convex analysis. (Juan Enrique Martínez-Legaz, Mathematical Reviews, February, 2023)Every chapter of the book has one section of exercises and one section of commentaries. These sections provide the reader with a lot of information and give him/her great benefits in self-learning. The book under review has many things to offer and, surely, it will play an important role in the development of convex analysis . The book is very useful for theoretical research and practical use. Thanks to the art of writing of the authors . (Nguyen Dong Yen, Journal of Global Optimization, Vol. 85, 2023)

1 Fundamentals
1(64)
1.1 Topological Spaces
1(31)
1.1.1 Definitions and Examples
1(4)
1.1.2 Topological Interior and Closure of Sets
5(2)
1.1.3 Continuity of Mappings
7(2)
1.1.4 Bases for Topologies
9(1)
1.1.5 Topologies Generated by Families of Mappings
10(2)
1.1.6 Product Topology and Quotient Topology
12(1)
1.1.7 Subspace Topology
13(2)
1.1.8 Separation Axioms
15(3)
1.1.9 Compactness
18(7)
1.1.10 Connectedness and Disconnectedness
25(3)
1.1.11 Net Convergence in Topological Spaces
28(4)
1.2 Topological Vector Spaces
32(17)
1.2.1 Basic Concepts in Topological Vector Spaces
32(7)
1.2.2 Weak Topology and Weak* Topology
39(7)
1.2.3 Quotient Spaces
46(3)
1.3 Some Fundamental Theorems of Functional Analysis
49(11)
1.3.1 Hahn-Banach Extension Theorem
50(4)
1.3.2 Baire Category Theorem
54(2)
1.3.3 Open Mapping Theorem
56(2)
1.3.4 Closed Graph Theorem and Uniform Boundedness Principle
58(2)
1.4 Exercises for
Chapter 1
60(3)
1.5 Commentaries to
Chapter 1
63(2)
2 Basic Theory Of Convexity
65(114)
2.1 Convexity of Sets
65(9)
2.1.1 Basic Definitions and Elementary Properties
65(4)
2.1.2 Operations on Convex Sets and Convex Hulls
69(5)
2.2 Cores, Minkowski Functions, and Seminorms
74(11)
2.2.1 Algebraic Interior and Linear Closure
74(4)
2.2.2 Minkowski Gauges
78(2)
2.2.3 Seminorms and Locally Convex Topologies
80(5)
2.3 Convex Separation Theorems
85(33)
2.3.1 Convex Separation in Vector Spaces
85(10)
2.3.2 Convex Separation in Topological Vector Spaces
95(7)
2.3.3 Convex Separation in Finite Dimensions
102(12)
2.3.4 Extreme Points of Convex Sets
114(4)
2.4 Convexity of Functions
118(29)
2.4.1 Descriptions and Properties of Convex Functions
118(6)
2.4.2 Convexity under Differentiability
124(4)
2.4.3 Operations Preserving Convexity of Functions
128(8)
2.4.4 Continuity of Convex Functions
136(6)
2.4.5 Lower Semicontinuity and Convexity
142(5)
2.5 Extended Relative Interiors in Infinite Dimensions
147(22)
2.5.1 Intrinsic Relative and Quasi-Relative Interiors
147(11)
2.5.2 Convex Separation via Extended Relative Interiors
158(5)
2.5.3 Extended Relative Interiors of Graphs and Epigraphs
163(6)
2.6 Exercises for
Chapter 2
169(4)
2.7 Commentaries to
Chapter 2
173(6)
3 Convex Generalized Differentiation
179(76)
3.1 The Normal Cone and Set Extremality
179(15)
3.1.1 Basic Definition and Normal Cone Properties
180(2)
3.1.2 Set Extremality and Convex Extremal Principle
182(3)
3.1.3 Normal Cone Intersection Rule in Topological Vector Spaces
185(5)
3.1.4 Normal Cone Intersection Rule in Finite Dimensions
190(4)
3.2 Coderivatives of Convex-Graph Mappings
194(7)
3.2.1 Coderivative Definition and Elementary Properties
195(1)
3.2.2 Coderivative Calculus in Topological Vector Spaces
196(4)
3.2.3 Coderivative Calculus in Finite Dimensions
200(1)
3.3 Subgradients of Convex Functions
201(31)
3.3.1 Basic Definitions and Examples
201(11)
3.3.2 Subdifferential Sum Rules
212(4)
3.3.3 Subdifferential Chain Rules
216(3)
3.3.4 Subdifferentiation of Maximum Functions
219(3)
3.3.5 Distance Functions and Their Subgradients
222(10)
3.4 Generalized Differentiation under Polyhedrality
232(13)
3.4.1 Polyhedral Convex Separation
232(7)
3.4.2 Polyhedral Normal Cone Intersection Rule
239(2)
3.4.3 Polyhedral Calculus for Coderivatives and Sub-differentials
241(4)
3.5 Exercises for
Chapter 3
245(4)
3.6 Commentaries to
Chapter 3
249(6)
4 Enhanced Calculus And Fenchel Duality
255(56)
4.1 Fenchel Conjugates
255(18)
4.1.1 Definitions, Examples, and Basic Properties
255(8)
4.1.2 Support Functions
263(4)
4.1.3 Conjugate Calculus
267(6)
4.2 Enhanced Calculus in Banach Spaces
273(5)
4.2.1 Support Functions of Set Intersections
273(2)
4.2.2 Refined Calculus Rules
275(3)
4.3 Directional Derivatives
278(5)
4.3.1 Definitions and Elementary Properties
279(1)
4.3.2 Relationships with Subgradients
280(3)
4.4 Subgradients of Supremum Functions
283(3)
4.4.1 Supremum of Convex Functions
283(2)
4.4.2 Subdifferential Formula for Supremum Functions
285(1)
4.5 Subgradients and Conjugates of Marginal Functions
286(6)
4.5.1 Computing Subgradients and Another Chain Rule
287(3)
4.5.2 Conjugate Calculations for Marginal Functions
290(2)
4.6 Fenchel Duality
292(10)
4.6.1 Fenchel Duality for Convex Composite Problems
292(6)
4.6.2 Duality Theorems via Generalized Relative Interiors
298(4)
4.7 Exercises for
Chapter 4
302(2)
4.8 Commentaries to
Chapter 4
304(7)
5 Variational Techniques And Further Subgradient Study
311(70)
5.1 Variational Principles and Convex Geometry
311(11)
5.1.1 Ekeland's Variational Principle and Related Results
312(3)
5.1.2 Convex Extremal Principles in Banach Spaces
315(3)
5.1.3 Density of e-Subgradients and Some Consequences
318(4)
5.2 Calculus Rules for E-Subgradients
322(6)
5.2.1 Exact Sum and Chain Rules for E-Subgradients
322(3)
5.2.2 Asymptotic e-Subdifferential Calculus
325(3)
5.3 Mean Value Theorems for Convex Functions
328(7)
5.3.1 Mean Value Theorem for Continuous Functions
328(2)
5.3.2 Approximate Mean Value Theorem
330(5)
5.4 Maximal Monotonicity of Subgradient Mappings
335(3)
5.5 Subdifferential Characterizations of Differentiability
338(15)
5.5.1 Gateaux and Frechet Differentiability
338(8)
5.5.2 Characterizations of Gateaux Differentiability
346(4)
5.5.3 Characterizations of Frechet Differentiability
350(3)
5.6 Generic Differentiability of Convex Functions
353(6)
5.6.1 Generic Gateaux Differentiability
354(2)
5.6.2 Generic Frechet Differentiability
356(3)
5.7 Spectral and Singular Functions in Convex Analysis
359(12)
5.7.1 Von Neumann Trace Inequality
359(3)
5.7.2 Spectral and Symmetric Functions
362(4)
5.7.3 Singular Functions and Their Subgradients
366(5)
5.8 Exercises for
Chapter 5
371(4)
5.9 Commentaries to
Chapter 5
375(6)
6 Miscellaneous Topics On Convexity
381(64)
6.1 Strong Convexity and Strong Smoothness
381(10)
6.1.1 Basic Definitions and Relationships
381(4)
6.1.2 Strong Convexity/Strong Smoothness via Derivatives
385(6)
6.2 Derivatives of Conjugates and Nesterov's Smoothing
391(7)
6.2.1 Differentiability of Conjugate Compositions
391(2)
6.2.2 Nesterov's Smoothing Techniques
393(5)
6.3 Convex Sets and Functions at Infinity
398(6)
6.3.1 Horizon Cones and Unboundedness
398(2)
6.3.2 Perspective and Horizon Functions
400(4)
6.4 Signed Distance Functions
404(6)
6.4.1 Basic Definition and Elementary Properties
405(2)
6.4.2 Lipschitz Continuity and Convexity
407(3)
6.5 Minimal Time Functions
410(14)
6.5.1 Minimal Time Functions with Constant Dynamics
410(6)
6.5.2 Subgradients of Minimal Time Functions
416(4)
6.5.3 Signed Minimal Time Functions
420(4)
6.6 Convex Geometry in Finite Dimensions
424(8)
6.6.1 Caratheodory Theorem on Convex Hulls
425(2)
6.6.2 Geometric Version of Farkas Lemma
427(3)
6.6.3 Radon and Helly Theorems on Set Intersections
430(2)
6.7 Approximations of Sets and Geometric Duality
432(3)
6.7.1 Full Duality between Tangent and Normal Cones
432(2)
6.7.2 Tangents and Normals for Polyhedral Sets
434(1)
6.8 Exercises for
Chapter 6
435(6)
6.9 Commentaries to
Chapter 6
441(4)
7 Convexified Lipschitzian Analysis
445(108)
7.1 Generalized Directional Derivatives
446(9)
7.1.1 Definitions and Relationships
446(6)
7.1.2 Properties of Extended Directional Derivatives
452(3)
7.2 Generalized Derivative and Subderivative Calculus
455(10)
7.2.1 Calculus Rules for Subderivatives
456(5)
7.2.2 Calculus of Generalized Directional Derivatives
461(4)
7.3 Directionally Generated Subdifferentials
465(15)
7.3.1 Basic Definitions and Some Properties
465(5)
7.3.2 Calculus Rules for Generalized Gradients
470(7)
7.3.3 Calculus of Contingent Subgradients
477(3)
7.4 Mean Value Theorems and More Calculus
480(6)
7.4.1 Mean Value Theorems for Lipschitzian Functions
480(3)
7.4.2 Additional Calculus Rules for Generalized Gradients
483(3)
7.5 Strict Differentiability and Generalized Gradients
486(7)
7.5.1 Notions of Strict Differentiability
487(3)
7.5.2 Single-Valuedness of Generalized Gradients
490(3)
7.6 Generalized Gradients in Finite Dimensions
493(7)
7.6.1 Rademacher Differentiability Theorem
494(1)
7.6.2 Gradient Representation of Generalized Gradients
495(2)
7.6.3 Generalized Gradients of Antiderivatives
497(3)
7.7 Subgradient Analysis of Distance Functions
500(14)
7.7.1 Regular and Limiting Subgradients of Lipschitzian Functions
500(5)
7.7.2 Regular and Limiting Subgradients of Distance Functions
505(5)
7.7.3 Subgradients of Convex Signed Distance Functions
510(4)
7.8 Differences of Convex Functions
514(22)
7.8.1 Continuous DC Functions
515(3)
7.8.2 The Mixing Property of DC Functions
518(7)
7.8.3 Locally DC Functions
525(7)
7.8.4 Subgradients and Conjugates of DC Functions
532(4)
7.9 Exercises for
Chapter 7
536(7)
7.10 Commentaries to
Chapter 7
543(10)
Glossary of Notation and Acronyms 553(4)
List of Figures 557(2)
References 559(18)
Subject Index 577
Boris Mordukhovich is Distinguished University Professor of Mathematics at Wayne State University. He has more than 500 publications including several monographs. Among his best known achievements are the introduction and development of powerful constructions of generalized differentiation and their applications to broad classes of problems in variational analysis, optimization, equilibrium, control, economics, engineering, and other fields. Mordukhovich is a SIAM Fellow, an AMS Fellow, and a recipient of many international awards and honors including Doctor Honoris Causa degrees from six universities over the world. He is a Highly Cited Researcher in Mathematics.  His research has been supported by continued grants from the National Science Foundations and the Air Force Office of Scientific Research. 







Nguyen Mau Nam is a Professor of Mathematics at Portland State University. He has published more than 55 research articles and one book in convex analysis withapplications to optimization theory and numerical algorithms. He has received several awards for his research including a best paper award by Journal of Global Optimization in 2021 and the Columbia-Willamette Chapter of Sigma Xi Outstanding Researcher Award in Mathematical Sciences in 2015. His research was supported by grants from the National Science Foundation, the Simons Foundation, and Portland State University.
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