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# Fundamentals of Convex Analysis

### Lecture Notes in Artificial Intelligence

Paperback Published: September 2001
ISBN: 9783540422051
Number Of Pages: 259

### Paperback

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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-

#### Industry Reviews

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 fur 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)

 Preface p. V Introduction: Notation, Elementary Results p. 1 Some Facts About Lower and Upper Bounds p. 1 The Set of ExtendedReal Numbers p. 5 Linear and Bilinear Algebra p. 6 Differentiationin a Euclidean Space p. 9 Set-Valued Analysis p. 12 Recalls on Convex Functions of the Real Variable p. 14 Exercises p. 16 Convex Sets p. 19 Generalities p. 19 Definition and First Examples p. 19 Convexity-PreservingOperationsonSets p. 22 ConvexCombinationsandConvexHulls p. 26 ClosedConvexSetsandHulls p. 31 ConvexSetsAttachedtoaConvexSet p. 33 TheRelativeInterior p. 33 TheAsymptoticCone p. 39 ExtremePoints p. 41 Exposed Faces p. 43 ProjectionontoClosedConvexSets p. 46 TheProjectionOperator p. 46 ProjectionontoaClosedConvexCone p. 49 Separation and Applications p. 51 SeparationBetweenConvexSets p. 51 First Consequences of the Separation Properties p. 54 Existence of Supporting Hyperplanes p. 54 Outer Description of Closed ConvexSets p. 55 Proof of Minkowski's Theorem p. 57 Bipolar of a ConvexCone p. 57 The Lemma of Minkowski-Farkas p. 58 ConicalApproximationsofConvexSets p. 62 ConvenientDefinitions of Tangent Cones p. 62 TheTangentandNormalConestoaConvexSet p. 65 SomePropertiesofTangentandNormalCones p. 67 Exercises p. 70 Convex Functions p. 73 Basic Definitions and Examples p. 73 The Definitions of a ConvexFunction p. 73 Special Convex Functions: Affinity and Closedness p. 76 Linear and Affine Functions p. 77 ClosedConvexFunctions p. 78 OuterConstructionofClosedConvexFunctions p. 80 FirstExamples p. 82 FunctionalOperationsPreservingConvexity p. 87 OperationsPreservingClosedness p. 87 Dilations and Perspectives of a Function p. 89 Infimal Convolution p. 92 Image of a Function Under a Linear Mapping p. 96 Convex Hull and Closed Convex Hull of a Function p. 98 Local and Global Behaviour of a Convex Function p. 102 Continuity Properties p. 102 Behaviour at Infinity p. 106 First- and Second-Order Differentiation p. 110 Differentiable ConvexFunctions p. 110 Nondifferentiable Convex Functions p. 113 Second-Order Differentiation p. 114 Exercises p. 117 Sublinearity and Support Functions p. 121 SublinearFunctions p. 123 Definitions and First Propertie p. 123 SomeExamples p. 127 TheConvexConeofAllClosedSublinearFunctions p. 131 The Support Function of a Nonempty Set p. 134 Definitions, Interpretations p. 134 BasicProperties p. 136 Examples p. 140 Correspondence Between Convex Sets and Sublinear Functions p. 143 The Fundamental Correspondence p. 143 Example: Norms and Their Duals, Polarity p. 146 Calculus with Support Functions p. 151 Example: Support Functions of Closed Convex Polyhedra p. 158 Exercises p. 161 Subdifferentials of Finite Convex Functions p. 163 The Subdifferential: Definitions and Interpretations p. 164 First Definition: Directional Derivatives p. 164 Second Definition: Minorizationby Affine Functions p. 167 GeometricConstructionsandInterpretations p. 169 Local Properties of the Subdifferential p. 173 First-OrderDevelopments p. 173 Minimality Conditions p. 176 Mean-ValueTheorems p. 177 FirstExamples p. 179 Calculus Rules with Subdifferentials p. 182 Positive Combinations of Functions p. 183 Pre-Composition with an Affine Mapping p. 184 Post-Composition with an Increasing Convex Function of Several Variables p. 185 Supremum of Convex Functions p. 187 Image of a Function Under a Linear Mapping p. 191 FurtherExamples p. 193 Largest Eigenvalue of a Symmetric Matrix p. 193 NestedOptimization p. 195 Best Approximation of a Continuous Function on a Compact Interval p. 197 The Subdifferential as a Multifunction p. 198 Monotonicity Properties of the Subdifferential p. 198 Continuity Properties of the Subdifferential p. 200 Subdifferentials and Limits of Subgradients p. 203 Exercises p. 204 Conjugacy in Convex Analysis p. 209 The Convex Conjugate of a Function p. 211 Definition and First Examples p. 211 Interpretations p. 214 FirstProperties p. 216 -Elementary Calculus Rules p. 216 -The Biconjugate of a Function p. 218 -ConjugacyandCoercivity p. 219 1.4 p. 220 Calculus Rules on the Conjugacy Operation p. 222 Image of a Function Under a Linear Mapping p. 222 Pre-Composition with an Affine Mapping p. 224 Sum of Two Functions p. 227 Infima and Suprema p. 229 Post-Composition with an Increasing Convex Function p. 231 Various Examples p. 233 The Cramer Transformation p. 234 The Conjugate of Convex Partially Quadratic Functions p. 234 PolyhedralFunctions p. 235 Differentiability of a Conjugate Function p. 237 First-Order Differentiability p. 238 Lipschitz Continuity of the Gradient Mapping p. 240 Exercises p. 241 Bibliographical Comments p. 245 The Founding Fathers of the Discipline p. 249 References p. 249 Index p. 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