Convex analysis

Title page......Page 1
Preface......Page 5
Introductory Remarks: a Guide for the Reader......Page 9
PART 1: BASIC CONCEPTS......Page 17
1. Affine Sets......Page 19
2. Convex Sets and Cones......Page 26
3. The Algebra of Convex Sets......Page 32
4. Convex Functions......Page 39
5. Functional Operations......Page 48
PART II: TOPOLOGICAL PROPERTIES......Page 57
6. Relative lnteriors of Convex Sets......Page 59
7. Closures of Convex Functions......Page 67
8. Recession Cones and Unboundedness......Page 76
9. Some Closedness Criteria......Page 88
10. Continuity of Convex Functions......Page 98
PART III: DUALITY CORRESPONDENCES......Page 109
11. Separation Theorems......Page 111
12. Conjugates of Convex Functions......Page 118
13. Support Functions......Page 128
14. Polars of Convex Sets......Page 137
15. Polars of Convex Functions......Page 144
16. Dual Operations......Page 156
PART IV: REPRESENTATION AND INEQUALITIES......Page 167
17. Carathéodory's Theorem......Page 169
18. Extreme Points and Faces of Convex Sets......Page 178
19. Polyhedral Convex Sets and Functions......Page 186
20. Some Applications of Polyhedral Convexity......Page 195
21. Helly's Theorem and Systems of Inequalities......Page 201
22. Linear Inequalities......Page 214
PART V; DlFFERENTlAL THEORY......Page 227
23. Directional Derivatives and Subgradients......Page 229
24. D ifferentiai Continuity and Monotonicity......Page 243
25. Differentiability of Convex Functions......Page 257
26. The Legendre Transformation......Page 267
PART VI: CONSTRAINED EXTREMUM PROBLEMS......Page 277
27. The Minimum of a Convex Function......Page 279
28. Ordinary Convex Programs and Lagrange Multipliers......Page 289
29. Bifunctions and Generalized Convex Programs......Page 307
30. Adjoint Bifunctions and Dual Programs......Page 323
31. Fenchel's Duality Theorem......Page 343
32. The Maximum of a Convex Function......Page 358
PART VII: SADDLE-FUNCTIONS AND MINIMAX THEORY......Page 363
33. Saddle-Functions......Page 365
34. Closures and Equivalence Classes......Page 375
35. Continuity and Differentiability of Saddle-functions......Page 386
36. Minimax Problems......Page 395
37. Conjugate Saddle-functions and Minimax Theorems......Page 404
PART VIII: CONVEX ALGEBRA......Page 415
38. The Algebra of Bifunctions......Page 417
39. Convex Processes......Page 429
Comments and References......Page 441
Bibliography......Page 449
Index......Page 463