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Fundamentals of Convex Analysis : Lecture Notes in Artificial Intelligence - Jean-Baptiste Hiriart-Urruty

Fundamentals of Convex Analysis

Lecture Notes in Artificial Intelligence

Paperback

Published: September 2001
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Foreword 0. Introduction: Notation, Elementary Results 1 Come Facts About Lower and Upper Bounds 2 The Set of Extended Real Numbers 3 Linear and Bilinear Algebra 4 Differentiation in a Euclidean Space 5 Set-Valued Analysis 6 Recalls on Convex Functions of the Real Variable Exercises A. Convex Sets 1. Generalities 1.1 Definitions and First Examples 1.2 Convexity-Preserving Operations on Sets 1.3 Convex Combinations and Convex Hulls 1.4 Closed Convex Sets and Hulls 2. Convex Sets Attached to a Convex Set 1.1 The Relative Interior 2.2 The Asymptotic Cone 2.3 Extreme Points 2.4 Exposed Faces 3. Projection onto Closed Convex Sets 3.1 The Projection Operator 3.2 Projection onto a Closed Convex Cone 4. Separation and Applications 4.1 Separation Between Convex Sets 4.2 First Consequences of the Separation Properties - Existence of Supporting Hyperplanes - Outer Description of Closed Convex Sets - Proof of Minkowski''s Theorem - Bipolar of a Convex Cone 4.3 The Lemma of Minkowski-Farkas 5. Conical Approximations of Convex Sets 5.1 Convenient Definitions of Tangent Cones 5.2 The Tangent and Normal Cones to a Convex Set 5.3 Some Properties of Tangent and Normal Cones Exercises B. Convex Functions 1. Basic Definitions and Examples 1.1 The Definitions of a Convex Function 1.2 Special Convex Functions: Affinity and Closedness - Linear and Affine Functions - Closed Convex Functions - Outer Construction of Closed Convex Functions 1.3 First Examples 2. Functional Operations Preserving Convexity 2.1 Operations Preserving Closedness 2.2 Dilations and Perspectives of a Function 2.3 Infimal Convolution 2.4 Image of a Functions Under a Linear Mapping 2.5 Convex Hull and Closed Convex Hull of a Function 3. Local and Global Behaviour of a Convex Function 3.1 Continuity Properties 3.2 Behaviour at Infinity 4. Fist- and Second-Order Differentiation 4.1 Differentiable Convex Functions 4.2 Nondifferentiable Convex Functions 4.3 Second-Order Differentiation Exercises C. Sublinearity and Support Functions 1. Sublinear Functions 1.1 Definitions and First Properties 1.2 Some Examples 1.3 The Convex Cone of All closed Sublinear Functions 2. The Support Function of a Nonempty Set 2.1 Definitions, Interpretations 2.2 Basic Properties 2.3 Examples 3. Correspondence Between Convex Sets and Sublinear Functions 3.1 The Fundamental Correspondence 3.2 Example: Norms and Their Duals, Polarity 3.3 Calculus with Support Functions 3.4 Example: Support Functions of Closed Convex Polyhedra Exercises D. Subdifferentials of Finite Convex Functions 1. The Subdifferential: Definitions and Interpretations 1.1 First Definition: Directional Derivatives 1.2 Second Definition: Minorization by Affine Functions 1.3 Geometric Constructions and Interpretations 2. Local Properties of the Subdifferential 2.1 First-Order Developments 2.2 Minimality conditions 2.3 Mean-Value Theorems 3. First Examples 4. Calculus Rules with Subdifferentials 4.1 Positive combinations of Functions 4.2 Pre-Compositions with an Affine Mapping 4.3 Post-composition with an Increasing Convex Functions of Several Variables 4.4 Supremum of Convex Functions 4.5 Image of a Functions Under a Linear Mapping 5. Further Examples 5.1 Largest Eigenvaule of a Symmetric Matrix 5.2 Nested Optimization 5.3 Best Approximation of a Continuous Function on a Compact Interval 6. The Subdifferential as a Multifunction 6.1 Monotonicity Properties of Subdifferential 6.2 Continuity Properties of the Subdifferential 6.3 Subdifferentials and Limits of Subgradients Exercises E. Conjugacy in Convex Analysis 1. The Convex Conjugate of a Function 1.1 Definition and First Examples 1.2 Interpretations 1.3 First Properties - Elementary Calculus Rules - The Biconjugate of a Function - Conjugacy and Coercivity 1.4 Subdifferntials of Extended-Valued Functions 2. Calculus Rules on the Conjugacy Operation 2.1 Image of a Function Under a Linear Mapping 2.2 Pre-Composition with an Affine Mapping 2.3 Sum of Two Functions 2.4 Infima and Suprema 2.5 Post-

From the reviews of the first edition:

..."This book is an abridged version of the book "Convex Analysis and Minimization Algorithms" (shortly CAMA) written in two volumes by the same authors... . The authors have extracted from CAMA Chapters III-VI and X, containing the fundamentals of convex analysis, deleting material seemed too advanced for an introduction, or too closely attached to numerical algorithms. Each Chapter is presented as a "lesson" treating a given subject in its entirety, completed by numerous examples and figures. So, this new version becomes a good book for learning and teaching of convex analysis in finite dimensions...."

S. Mititelu in "Zentralblatt fr  Mathematik und ihre Grenzgebiete," 2002

"I believe that the book under review will become the standard text doing much to implement the type of course Victor Klee was advocating and covering as it does the considerable recent development of the subject. If you are looking for a well-designed text for a course on convex analysis, preliminary to one on optimization or nonlinear analysis then this is the one which will certainly be a standard for many years." (John Giles, The Australian Mathematical Society Gazette, Vol. 29 (2), 2002)

Prefacep. V
Introduction: Notation, Elementary Resultsp. 1
Some Facts About Lower and Upper Boundsp. 1
The Set of ExtendedReal Numbersp. 5
Linear and Bilinear Algebrap. 6
Differentiationin a Euclidean Spacep. 9
Set-Valued Analysisp. 12
Recalls on Convex Functions of the Real Variablep. 14
Exercisesp. 16
Convex Setsp. 19
Generalitiesp. 19
Definition and First Examplesp. 19
Convexity-PreservingOperationsonSetsp. 22
ConvexCombinationsandConvexHullsp. 26
ClosedConvexSetsandHullsp. 31
ConvexSetsAttachedtoaConvexSetp. 33
TheRelativeInteriorp. 33
TheAsymptoticConep. 39
ExtremePointsp. 41
Exposed Facesp. 43
ProjectionontoClosedConvexSetsp. 46
TheProjectionOperatorp. 46
ProjectionontoaClosedConvexConep. 49
Separation and Applicationsp. 51
SeparationBetweenConvexSetsp. 51
First Consequences of the Separation Propertiesp. 54
Existence of Supporting Hyperplanesp. 54
Outer Description of Closed ConvexSetsp. 55
Proof of Minkowski's Theoremp. 57
Bipolar of a ConvexConep. 57
The Lemma of Minkowski-Farkasp. 58
ConicalApproximationsofConvexSetsp. 62
ConvenientDefinitions of Tangent Conesp. 62
TheTangentandNormalConestoaConvexSetp. 65
SomePropertiesofTangentandNormalConesp. 67
Exercisesp. 70
Convex Functionsp. 73
Basic Definitions and Examplesp. 73
The Definitions of a ConvexFunctionp. 73
Special Convex Functions: Affinity and Closednessp. 76
Linear and Affine Functionsp. 77
ClosedConvexFunctionsp. 78
OuterConstructionofClosedConvexFunctionsp. 80
FirstExamplesp. 82
FunctionalOperationsPreservingConvexityp. 87
OperationsPreservingClosednessp. 87
Dilations and Perspectives of a Functionp. 89
Infimal Convolutionp. 92
Image of a Function Under a Linear Mappingp. 96
Convex Hull and Closed Convex Hull of a Functionp. 98
Local and Global Behaviour of a Convex Functionp. 102
Continuity Propertiesp. 102
Behaviour at Infinityp. 106
First- and Second-Order Differentiationp. 110
Differentiable ConvexFunctionsp. 110
Nondifferentiable Convex Functionsp. 113
Second-Order Differentiationp. 114
Exercisesp. 117
Sublinearity and Support Functionsp. 121
SublinearFunctionsp. 123
Definitions and First Propertiep. 123
SomeExamplesp. 127
TheConvexConeofAllClosedSublinearFunctionsp. 131
The Support Function of a Nonempty Setp. 134
Definitions, Interpretationsp. 134
BasicPropertiesp. 136
Examplesp. 140
Correspondence Between Convex Sets and Sublinear Functionsp. 143
The Fundamental Correspondencep. 143
Example: Norms and Their Duals, Polarityp. 146
Calculus with Support Functionsp. 151
Example: Support Functions of Closed Convex Polyhedrap. 158
Exercisesp. 161
Subdifferentials of Finite Convex Functionsp. 163
The Subdifferential: Definitions and Interpretationsp. 164
First Definition: Directional Derivativesp. 164
Second Definition: Minorizationby Affine Functionsp. 167
GeometricConstructionsandInterpretationsp. 169
Local Properties of the Subdifferentialp. 173
First-OrderDevelopmentsp. 173
Minimality Conditionsp. 176
Mean-ValueTheoremsp. 177
FirstExamplesp. 179
Calculus Rules with Subdifferentialsp. 182
Positive Combinations of Functionsp. 183
Pre-Composition with an Affine Mappingp. 184
Post-Composition with an Increasing Convex Function of Several Variablesp. 185
Supremum of Convex Functionsp. 187
Image of a Function Under a Linear Mappingp. 191
FurtherExamplesp. 193
Largest Eigenvalue of a Symmetric Matrixp. 193
NestedOptimizationp. 195
Best Approximation of a Continuous Function on a Compact Intervalp. 197
The Subdifferential as a Multifunctionp. 198
Monotonicity Properties of the Subdifferentialp. 198
Continuity Properties of the Subdifferentialp. 200
Subdifferentials and Limits of Subgradientsp. 203
Exercisesp. 204
Conjugacy in Convex Analysisp. 209
The Convex Conjugate of a Functionp. 211
Definition and First Examplesp. 211
Interpretationsp. 214
FirstPropertiesp. 216
-Elementary Calculus Rulesp. 216
-The Biconjugate of a Functionp. 218
-ConjugacyandCoercivityp. 219
1.4p. 220
Calculus Rules on the Conjugacy Operationp. 222
Image of a Function Under a Linear Mappingp. 222
Pre-Composition with an Affine Mappingp. 224
Sum of Two Functionsp. 227
Infima and Supremap. 229
Post-Composition with an Increasing Convex Functionp. 231
Various Examplesp. 233
The Cramer Transformationp. 234
The Conjugate of Convex Partially Quadratic Functionsp. 234
PolyhedralFunctionsp. 235
Differentiability of a Conjugate Functionp. 237
First-Order Differentiabilityp. 238
Lipschitz Continuity of the Gradient Mappingp. 240
Exercisesp. 241
Bibliographical Commentsp. 245
The Founding Fathers of the Disciplinep. 249
Referencesp. 249
Indexp. 256
Table of Contents provided by Publisher. All Rights Reserved.

ISBN: 9783540422051
ISBN-10: 3540422056
Series: Lecture Notes in Artificial Intelligence
Audience: General
Format: Paperback
Language: English
Number Of Pages: 259
Published: September 2001
Publisher: Springer-Verlag Berlin and Heidelberg Gmbh & Co. Kg
Country of Publication: DE
Dimensions (cm): 23.39 x 15.6  x 1.45
Weight (kg): 0.39
Edition Type: Abridged