Fundamentals of Convex Analysis

Couverture
Springer Science & Business Media, 21 avr. 2004 - 259 pages
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.
 

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Table des matières

A Convex Sets
19
12 ConvexityPreserving Operations on Sets
22
13 Convex Combinations and Convex Hulls
26
14 Closed Convex Sets and Hulls
31
2 Convex Sets Attached to a Convex Set
33
22 The Asymptotic Cone
39
23 Extreme Points
41
24 Exposed Faces
43
Support Functions of Closed Convex Polyhedra
158
Exercises
161
D Subdifferentials of Finite Convex Functions
163
Definitions and Interpretations
164
Minorization by Affine Functions
167
13 Geometric Constructions and Interpretations
169
2 Local Properties of the Subdifferential
173
22 Minimality Conditions
177

3 Projection onto Closed Convex Sets
46
32 Projection onto a Closed Convex Cone
49
4 Separation and Applications
51
42 First Consequences of the Separation Properties
54
b Outer Description of Closed Convex Sets
55
c Proof of Minkowskis Theorem
57
43 The Lemma of MinkowskiFarkas
58
5 Conical Approximations of Convex Sets
62
52 The Tangent and Normal Cones to a Convex Set
65
53 Some Properties of Tangent and Normal Cones
68
Exercises
70
B Convex Functions
73
Affinity and Closedness
76
a Linear and Affine Functions
77
b Closed Convex Functions
78
c Outer Construction of Closed Convex Functions
80
13 First Examples
82
2 Functional Operations Preserving Convexity
87
22 Dilations and Perspectives of a Function
89
23 Infimal Convolution
92
24 Image of a Function Under a Linear Mapping
96
25 Convex Hull and Closed Convex Hull of a Function
98
3 Local and Global Behaviour of a Convex Function
102
32 Behaviour at Infinity
106
4 First and SecondOrder Differentiation
110
42 Nondifferentiable Convex Functions
114
43 SecondOrder Differentiation
115
Exercises
117
C Sublinearity and Support Functions
121
1 Sublinear Functions
123
12 Some Examples
127
13 The Convex Cone of All Closed Sublinear Functions
131
2 The Support Function of a Nonempty Set
134
22 Basic Properties
136
23 Examples
140
3 The Isomorphism Between Closed Convex Sets and Closed Sublinear Functions
143
Norms and Their Duals Polarity
146
33 Calculus with Support Functions
151
23 MeanValue Theorems
178
3 First Examples
180
4 Calculus Rules with Subdifferentials
183
42 PreComposition with an Affine Mapping
184
43 PostComposition with an Increasing Convex Function of Several Variables
185
44 Supremum of Convex Functions
188
45 Image of a Function Under a Linear Mapping
191
5 Further Examples
194
52 Nested Optimization
196
53 Best Approximation of a Continuous Function on a Compact Interval
198
6 The Subdifferential as a Multifunction
199
62 Continuity Properties of the Subdifferential
201
63 Subdifferentials and Limits of Subgradients
204
Exercises
205
E Conjugacy in Convex Analysis
209
1 The Convex Conjugate of a Function
211
12 Interpretations
214
13 First Properties
216
b The Kiconjugate of a Function
218
c Conjugacy and Coercivity
219
14 Subdifferentials of ExtendedValued Functions
220
2 Calculus Rules on the Conjugacy Operation
222
22 PreComposition with an Affine Mapping
224
23 Sum of Two Functions
227
24 Infima and Suprema
229
25 PostComposition with an Increasing Convex Function
231
3 Various Examples
233
31 The Cramer Transformation
234
33 Polyhedral Functions
235
4 Differentiability of a Conjugate Function
237
41 FirstOrder Differentiability
238
42 Lipschitz Continuity of the Gradient Mapping
240
Exercises
241
Bibliographical Comments
245
The Founding Fathers of the Discipline
249
References
251
Index
253
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