# Fundamentals of Convex Analysis

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 Droits d'auteur

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Page vi - Everything should be made as simple as possible, but not simpler".