## Fundamentals of Convex AnalysisSpringer 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 |

251 | |

253 | |

### Autres éditions - Tout afficher

Fundamentals of Convex Analysis Jean-Baptiste Hiriart-Urruty,Claude Lemaréchal Aucun aperçu disponible |

### Expressions et termes fréquents

affine function affine hull affine manifold arbitrary as(d assumption bounded calculus rules closed convex cone closed convex function closed convex hull closed convex set closed sublinear function closure compact set compute conjugate consider contains Conv Rn Conversely convex analysis convex combination convex cone convex hull Corollary corresponding defined denote df(x epi f epigraph equivalent Euclidean Example exposed face extreme point f is convex finite function f geometric given gradient half-spaces implies inequality infimal convolution infimum intersection Lemma Let f Lipschitz matrix minimization nonempty closed convex nonnegative norm normal cone notation obtain optimal orthogonal pc(x polar polyhedral positive homogeneity Proof Proposition prove quadratic relative interior Remark resp result satisfying scalar product semi-continuous Show space subdifferential subgradient sublevel-set sublinear functions subspace support function supremum symmetric tangent cone Tc(x Theorem vector

### Fréquemment cités

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