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