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: |