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# Fundamentals of Convex Analysis

## byJean-Baptiste Hiriart-Urruty, Claude Lemaréchal

### Paperback | April 21, 2004

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This book is an abridged version of our two-volume opus Convex Analysis and Minimization Algorithms [18], about which we have received very positive feedback from users, readers, lecturers ever since it was published - by Springer-Verlag in 1993. Its pedagogical qualities were particularly appreciated, in the combination with a rather advanced technical material. Now [18] hasa dual but clearly defined nature: - an introduction to the basic concepts in convex analysis, - a study of convex minimization problems (with an emphasis on numerical al- rithms), and insists on their mutual interpenetration. It is our feeling that the above basic introduction is much needed in the scientific community. This is the motivation for the present edition, our intention being to create a tool useful to teach convex anal ysis. We have thus extracted from [18] its "backbone" devoted to convex analysis, namely ChapsIII-VI and X. Apart from some local improvements, the present text is mostly a copy of the corresponding chapters. The main difference is that we have deleted material deemed too advanced for an introduction, or too closely attached to numerical algorithms. Further, we have included exercises, whose degree of difficulty is suggested by 0, I or 2 stars *. Finally, the index has been considerably enriched. Just as in [18], each chapter is presented as a "lesson", in the sense of our old masters, treating of a given subject in its entirety.

### Details & Specs

Title:Fundamentals of Convex AnalysisFormat:PaperbackPublished:April 21, 2004Publisher:Springer-Verlag/Sci-Tech/TradeLanguage:English

The following ISBNs are associated with this title:

ISBN - 10:3540422056

ISBN - 13:9783540422051

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Table of Contents

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-Composition with an Increasing Convex Function 3. Various Examples 3.1 The Cramer Transformation 3.2 The Conjugate of convex Partially Quadratic Functions 3.3 Polyhedral Functions 4. Differentiability of a Conjugate Function 4.1 Fist-Order Differentiability 4.2 Lipschitz Continuity of the Gradient Mapping Exercises Bibliographical Comments References Index

Editorial 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 für 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)