Fundamentals of Convex Analysis,9783540422051

Fundamentals of Convex Analysis

by ; ; ; ; ;
ISBN13: 9783540422051
ISBN10: 3540422056
Format: Hardcover
Pub. Date: 2001-09-01
Publisher(s): Springer Verlag
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Summary

This book is an abridged version of the two volumes "Convex Analysis and Minimization Algorithms I and II" (Grundlehren der mathematischen Wissenschaften Vol. 305 and 306), which presented an introduction to the basic concepts in convex analysis and a study of convex minimization problems. The "backbone" of both volumes was extracted, some material deleted that was deemed too advanced for an introduction, or too closely related to numerical algorithms. Some exercises were included and finally the index has been considerably enriched. The main motivation of the authors was to "light the entrance" of the monument Convex Analysis. This book is not a reference book to be kept on the shelf by experts who already know the building and can find their way through it; it is far more a book for the purpose of learning and teaching.

Table of Contents

Preface v
Introduction: Notation, Elementary Results
1(18)
Some Facts About Lower and Upper Bounds
1(4)
The Set of Extended Real Numbers
5(1)
Linear and Bilinear Algebra
6(3)
Differentiation in a Euclidean Space
9(3)
Set-Valued Analysis
12(2)
Recalls on Convex Functions of the Real Variable
14(5)
Exercises
16(3)
Convex Sets
19(54)
Generalities
19(14)
Definition and First Examples
19(3)
Convexity-Preserving Operations on Sets
22(4)
Convex Combinations and Convex Hulls
26(5)
Closed Convex Sets and Hulls
31(2)
Convex Sets Attached to a Convex Set
33(13)
The Relative Interior
33(6)
The Asymptotic Cone
39(2)
Extreme Points
41(2)
Exposed Faces
43(3)
Projection onto Closed Convex Sets
46(5)
The Projection Operator
46(3)
Projection onto a Closed Convex Cone
49(2)
Separation and Applications
51(11)
Separation Between Convex Sets
51(3)
First Consequences of the Separation Properties
54(1)
Existence of Supporting Hyperplanes
54(1)
Outer Description of Closed Convex Sets
55(2)
Proof of Minkowski's Theorem
57(1)
Bipolar of a Convex Cone
57(1)
The Lemma of Minkowski-Farkas
58(4)
Conical Approximations of Convex Sets
62(11)
Convenient Definitions of Tangent Cones
62(3)
The Tangent and Normal Cones to a Convex Set
65(3)
Some Properties of Tangent and Normal Cones
68(2)
Exercises
70(3)
Convex Functions
73(48)
Basic Definitions and Examples
73(14)
The Definitions of a Convex Function
73(3)
Special Convex Functions: Affinity and Closedness
76(1)
Linear and Affine Functions
77(1)
Closed Convex Functions
78(2)
Outer Construction of Closed Convex Functions
80(2)
First Examples
82(5)
Functional Operations Preserving Convexity
87(15)
Operations Preserving Closedness
87(2)
Dilations and Perspectives of a Function
89(3)
Infimal Convolution
92(4)
Image of a Function Under a Linear Mapping
96(2)
Convex Hull and Closed Convex Hull of a Function
98(4)
Local and Global Behaviour of a Convex Function
102(8)
Continuity Properties
102(4)
Behaviour at Infinity
106(4)
First-and Second-Order Differentiation
110(11)
Differentiable Convex Functions
110(4)
Nondifferentiable Convex Functions
114(1)
Second-Order Differentiation
115(2)
Exercises
117(4)
Sublinearity and Support Functions
121(42)
Sublinear Functions
123(11)
Definitions and First Properties
123(4)
Some Examples
127(4)
The Convex Cone of All Closed Sublinear Functions
131(3)
The Support Function of a Nonempty Set
134(9)
Definitions, Interpretations
134(2)
Basic Properties
136(4)
Examples
140(3)
Correspondence Between Convex Sets and Sublinear Functions
143(20)
The Fundamental Correspondence
143(3)
Example: Norms and Their Duals, Polarity
146(5)
Calculus with Support Functions
151(7)
Example: Support Functions of Closed Convex Polyhedra
158(3)
Exercises
161(2)
Subdifferentials of Finite Convex Functions
163(46)
The Subdifferential: Definitions and Interpretations
164(9)
First Definition: Directional Derivatives
164(3)
Second Definition: Minorization by Affine Functions
167(2)
Geometric Constructions and Interpretations
169(4)
Local Properties of the Subdifferential
173(7)
First-Order Developments
173(4)
Minimality Conditions
177(1)
Mean-Value Theorems
178(2)
First Examples
180(3)
Calculus Rules with Subdifferentials
183(11)
Positive Combinations of Functions
183(1)
Pre-Composition with an Affine Mapping
184(1)
Post-Composition with an Increasing Convex Function of Several Variables
185(3)
Supremum of Convex Functions
188(3)
Image of a Function Under a Linear Mapping
191(3)
Further Examples
194(5)
Largest Eigenvalue of a Symmetric Matrix
194(2)
Nested Optimization
196(2)
Best Approximation of a Continuous Function on a Compact Interval
198(1)
The Subdifferential as a Multifunction
199(10)
Monotonicity Properties of the Subdifferential
199(2)
Continuity Properties of the Subdifferential
201(3)
Subdifferentials and Limits of Subgradients
204(1)
Exercises
205(4)
Conjugacy in Convex Analysis
209(36)
The Convex Conjugate of a Function
211(11)
Definition and First Examples
211(3)
Interpretations
214(2)
First Properties
216(1)
Elementary Calculus Rules
216(2)
The Biconjugate of a Function
218(1)
Conjugacy and Coercivity
219(1)
Subdifferentials of Extended-Valued Functions
220(2)
Calculus Rules on the Conjugacy Operation
222(11)
Image of a Function Under a Linear Mapping
222(2)
Pre-Composition with an Affine Mapping
224(3)
Sum of Two Functions
227(2)
Infima and Suprema
229(2)
Post-Composition with an Increasing Convex Function
231(2)
Various Examples
233(4)
The Cramer Transformation
234(1)
The Conjugate of Convex Partially Quadratic Functions
234(1)
Polyhedral Functions
235(2)
Differentiability of a Conjugate Function
237(8)
First-Order Differentiability
238(2)
Lipschitz Continuity of the Gradient Mapping
240(1)
Exercises
241(4)
Bibliographical Comments 245(4)
The Founding Fathers of the Discipline 249(2)
References 251(2)
Index 253

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