Fundamentals of convex analysis

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Fundamentals of convex analysis / Jean-Baptiste Hiriart-Urruty, Claude Lemaréchal

Format:

Book

Published:

Berlin : Springer, c2001

Description:

x, 259 p. ; 24 cm

Series:

Grundlehren text editions / editors, A. Chenciner ... [et al.] ; managing editors, M. Berger, J. Coates, S.R.S. Varadhan <BA53820480>

Authors:

ISBN:

9783540422051 [3540422056]

NCID:

BA53820593

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Bibliography Details

Note:

"Abridged version of our two-volume opus Convex analysis and minimization algorithms ... published ... in 1993" -- Preface
Includes bibliographical references and index

Subject:

Convex functions; Convex sets

Language:

English

Table of Contents:

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