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Tables of Contents for Convex Analysis
Chapter/Section Title
Page #
Page Count
Preface
vii
4
Introductory Remarks: a Guide for the Reader
xi
PART I: BASIC CONCEPTS
3
40
1. Affine Sets
3
7
2. Convex Sets and Cones
10
6
3. The Algebra of Convex Sets
16
7
4. Convex Functions
23
9
5. Functional Operations
32
11
PART II: TOPOLOGICAL PROPERTIES
43
52
6. Relative Interiors of Convex Sets
43
8
7. Closures of Convex Functions
51
9
8. Recession Cones and Unboundedness
60
12
9. Some Closedness Criteria
72
10
10. Continuity of Convex Functions
82
13
PART III: DUALITY CORRESPONDENCES
95
58
11. Separation Theorems
95
7
12. Conjugates of Convex Functions
102
10
13. Support Functions
112
9
14. Polars of Convex Sets
121
7
15. Polars of Convex Functions
128
12
16. Dual Operations
140
13
PART IV: REPRESENTATION AND INEQUALITIES
153
60
17. Caratheodory's Theorem
153
9
18. Extreme Points and Faces of Convex Sets
162
8
19. Polyhedral Convex Sets and Functions
170
9
20. Some Applications of Polyhedral Convexity
179
6
21. Helly's Theorem and Systems of Inequalities
185
13
22. Linear Inequalities
198
15
PART V: DIFFERENTIAL THEORY
213
50
23. Directional Derivatives and Subgradients
213
14
24. Differntial Continuity and Monotonicity
227
14
25. Differentiability of Convex Functions
241
10
26. The Legendre Transformation
251
12
PART VI: CONTRAINED EXTREMUM PROBLEMS
263
86
27. The Minimum of a Convex Function
263
10
28. Ordinary Convex Programs and Legrange Multipliers
273
18
29. Bifunctions and Generalized Convex Programs
291
16
30. Adjoint Bifunctions and Dual Programs
307
20
31. Fenchel's Duality Theorem
327
15
32. The Maximum of a Convex Function
342
7
PART VII: SADDLE-FUNCTIONS AND MINIMAX THEORY
349
52
33. Saddle-Functions
349
10
34. Closures and Equivalence Classes
359
11
35. Continuity and Differentiability of Saddle-functions
370
9
36. Minimax Problems
379
9
37. Conjugate Saddle-functions and Minimax Theorems
388
13
PART VIII: CONVEX ALGEBRA
401
24
38. The Algebra of Bifunctions
401
12
39. Convex Processes
413
12
Comments and References
425
8
Bibliography
433
14
Index
447