Fundamentals of Convex Analysis

ISBN-10: 3540422056
ISBN-13: 9783540422051
Edition: 2001
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Description: This book is an abridged version of the two volumes "Convex Analysis and Minimization Algorithms I and II" (Grundlehren der mathematischen Wissenschaften Vol. 305 and 306), which presented an introduction to the basic concepts in convex analysis and  More...

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Book details

Copyright year: 2001
Publisher: Springer
Publication date: 4/21/2004
Binding: Paperback
Pages: 259
Size: 6.00" wide x 9.00" long x 0.75" tall
Weight: 0.880
Language: English

This book is an abridged version of the two volumes "Convex Analysis and Minimization Algorithms I and II" (Grundlehren der mathematischen Wissenschaften Vol. 305 and 306), which presented an introduction to the basic concepts in convex analysis and a study of convex minimization problems. The "backbone" of both volumes was extracted, some material deleted that was deemed too advanced for an introduction, or too closely related to numerical algorithms. Some exercises were included and finally the index has been considerably enriched. The main motivation of the authors was to "light the entrance" of the monument Convex Analysis. This book is not a reference book to be kept on the shelf by experts who already know the building and can find their way through it; it is far more a book for the purpose of learning and teaching.

Preface
Introduction: Notation, Elementary Results
Some Facts About Lower and Upper Bounds
The Set of ExtendedReal Numbers
Linear and Bilinear Algebra
Differentiationin a Euclidean Space
Set-Valued Analysis
Recalls on Convex Functions of the Real Variable
Exercises
Convex Sets
Generalities
Definition and First Examples
Convexity-PreservingOperationsonSets
ConvexCombinationsandConvexHulls
ClosedConvexSetsandHulls
ConvexSetsAttachedtoaConvexSet
TheRelativeInterior
TheAsymptoticCone
ExtremePoints
Exposed Faces
ProjectionontoClosedConvexSets
TheProjectionOperator
ProjectionontoaClosedConvexCone
Separation and Applications
SeparationBetweenConvexSets
First Consequences of the Separation Properties
Existence of Supporting Hyperplanes
Outer Description of Closed ConvexSets
Proof of Minkowski's Theorem
Bipolar of a ConvexCone
The Lemma of Minkowski-Farkas
ConicalApproximationsofConvexSets
ConvenientDefinitions of Tangent Cones
TheTangentandNormalConestoaConvexSet
SomePropertiesofTangentandNormalCones
Exercises
Convex Functions
Basic Definitions and Examples
The Definitions of a ConvexFunction
Special Convex Functions: Affinity and Closedness
Linear and Affine Functions
ClosedConvexFunctions
OuterConstructionofClosedConvexFunctions
FirstExamples
FunctionalOperationsPreservingConvexity
OperationsPreservingClosedness
Dilations and Perspectives of a Function
Infimal Convolution
Image of a Function Under a Linear Mapping
Convex Hull and Closed Convex Hull of a Function
Local and Global Behaviour of a Convex Function
Continuity Properties
Behaviour at Infinity
First- and Second-Order Differentiation
Differentiable ConvexFunctions
Nondifferentiable Convex Functions
Second-Order Differentiation
Exercises
Sublinearity and Support Functions
SublinearFunctions
Definitions and First Propertie
SomeExamples
TheConvexConeofAllClosedSublinearFunctions
The Support Function of a Nonempty Set
Definitions, Interpretations
BasicProperties
Examples
Correspondence Between Convex Sets and Sublinear Functions
The Fundamental Correspondence
Example: Norms and Their Duals, Polarity
Calculus with Support Functions
Example: Support Functions of Closed Convex Polyhedra
Exercises
Subdifferentials of Finite Convex Functions
The Subdifferential: Definitions and Interpretations
First Definition: Directional Derivatives
Second Definition: Minorizationby Affine Functions
GeometricConstructionsandInterpretations
Local Properties of the Subdifferential
First-OrderDevelopments
Minimality Conditions
Mean-ValueTheorems
FirstExamples
Calculus Rules with Subdifferentials
Positive Combinations of Functions
Pre-Composition with an Affine Mapping
Post-Composition with an Increasing Convex Function of Several Variables
Supremum of Convex Functions
Image of a Function Under a Linear Mapping
FurtherExamples
Largest Eigenvalue of a Symmetric Matrix
NestedOptimization
Best Approximation of a Continuous Function on a Compact Interval
The Subdifferential as a Multifunction
Monotonicity Properties of the Subdifferential
Continuity Properties of the Subdifferential
Subdifferentials and Limits of Subgradients
Exercises
Conjugacy in Convex Analysis
The Convex Conjugate of a Function
Definition and First Examples
Interpretations
FirstProperties
-Elementary Calculus Rules
-The Biconjugate of a Function
-ConjugacyandCoercivity
1.4
Calculus Rules on the Conjugacy Operation
Image of a Function Under a Linear Mapping
Pre-Composition with an Affine Mapping
Sum of Two Functions
Infima and Suprema
Post-Composition with an Increasing Convex Function
Various Examples
The Cramer Transformation
The Conjugate of Convex Partially Quadratic Functions
PolyhedralFunctions
Differentiability of a Conjugate Function
First-Order Differentiability
Lipschitz Continuity of the Gradient Mapping
Exercises
Bibliographical Comments
The Founding Fathers of the Discipline
References
Index

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