| Preface |
| Introduction: Notation, Elementary Results / 0： |
| Some Facts About Lower and Upper Bounds / 1： |
| The Set of ExtendedReal Numbers / 2： |
| Linear and Bilinear Algebra / 3： |
| Differentiationin a Euclidean Space / 4： |
| Set-Valued Analysis / 5： |
| Recalls on Convex Functions of the Real Variable / 6： |
| Exercises |
| Convex Sets / A： |
| Generalities |
| Definition and First Examples / 1.1： |
| Convexity-PreservingOperationsonSets / 1.2： |
| ConvexCombinationsandConvexHulls / 1.3： |
| ClosedConvexSetsandHulls / 1.4： |
| ConvexSetsAttachedtoaConvexSet |
| TheRelativeInterior / 2.1： |
| TheAsymptoticCone / 2.2： |
| ExtremePoints / 2.3： |
| Exposed Faces / 2.4： |
| ProjectionontoClosedConvexSets |
| TheProjectionOperator / 3.1： |
| ProjectionontoaClosedConvexCone / 3.2： |
| Separation and Applications |
| SeparationBetweenConvexSets / 4.1： |
| First Consequences of the Separation Properties / 4.2： |
| Existence of Supporting Hyperplanes |
| Outer Description of Closed ConvexSets |
| Proof of Minkowski's Theorem |
| Bipolar of a ConvexCone |
| The Lemma of Minkowski-Farkas / 4.3： |
| ConicalApproximationsofConvexSets |
| ConvenientDefinitions of Tangent Cones / 5.1： |
| TheTangentandNormalConestoaConvexSet / 5.2： |
| SomePropertiesofTangentandNormalCones / 5.3： |
| Convex Functions / B： |
| 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 / 2.5： |
| 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 |
| Sublinearity and Support Functions / C： |
| 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 / 3.3： |
| Example: Support Functions of Closed Convex Polyhedra / 3.4： |
| Subdifferentials of Finite Convex Functions / D： |
| 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 |
| 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 / 4.4： |
| FurtherExamples / 4.5： |
| 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 / 6.1： |
| Continuity Properties of the Subdifferential / 6.2： |
| Subdifferentials and Limits of Subgradients / 6.3： |
| Conjugacy in Convex Analysis / E： |
| The Convex Conjugate of a Function |
| Interpretations |
| FirstProperties |
| -Elementary Calculus Rules |
| -The Biconjugate of a Function |
| -ConjugacyandCoercivity |
| Subdifferentials of Extended-Valued Functions |
| Calculus Rules on the Conjugacy Operation |
| 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 |
| Bibliographical Comments |
| The Founding Fathers of the Discipline |
| References |
| Index |
| Preface |
| Introduction: Notation, Elementary Results / 0： |
| Some Facts About Lower and Upper Bounds / 1： |
図書
