內容簡介
《凸分析(英文版)》內容簡介：Convexity has been increasingly important in recent years in the study of extremum problems in many areas of applied mathematics. The purpose of this book is to provide an exposition of the theory of convex sets and functions in which applications to extremum problems play the centralrole. Systems of inequalities, the minimum or maximum of a convex functionover a convex set, Lagrange multipliers, and minimax theorems are among the topics treated, as well as basic results about the structure of convex sets and the continuity and differentiability of convex functions and saddle-functions. Duality is emphasized throughout, particularly in the form of Fenchers conjugacy correspondence for convex functions.
目錄
Preface .
Introductory Remarks: a Guide for the Reader
PART l: BASIC CONCEPTS
1. Affine Sets
2. Convex Sets and Cones
3. The Algebra of Convex Sets
4. Convex Functions
5. Functional Operations
PART II: TOPOLOGICAL PROPERTIES
6. Relative Interiors of Convex Sels
7. Closures of Convex Functions
8. Recession Cones and Unboundedness
9. Some CIosedness Criteria
10. Continuity of Convex Functions
PART Ⅲ: DUALITY CORRESPONDENCES
11. Separation Theorems
12. Conjugates of Convex Functions
13. Support Furctions
14. Polars of Convex Sets
15. Polars of Convex Functions
16.Dual Operations
PART IV: REPRESENTATION AND INEQUALITIES
17. Carath6odory's Theorem
18. Extreme Points and Faces of Convex Sets
19. Polyhedral Convex Sets and Functions
20. Some Applications of Polyhedral Convexity
21.Helly's Theorem and Systems of Inequalities
22. Linear Inequalities
CONTENTS
PART V: DIFFERENTIAL THEORY
23. Directional Derivatives and Subgradients
24. Differential Continuity and Monotonicity
25. Differentiability of Convex Functions
26. The Legendre Transformation
PART VI: CONSTRAINED EXTREMUM PROBLEMS
27. The Minimum of a Convex Function
28. Ordinary Convex Programs and Lagrange Multipliers
29. Bifunctions and Generalized Convex Programs
30. Adjoint Bifunctions and Dual Programs
31. Fenchel's Duality Theorem
32. The Maximum of a Convex Function
PART VII:' SADDLE-FUNCTIONS AND MINIMAX THEORY
33. Saddle-Functions
34. Closures and Equivalence Classes
35. Continuity and Differentiability of Saddle-functions
36. Minimax Problems
37. Conjugate Saddle-functions and Minimax Theorems
PART VIII: CONVEX ALGEBRA
38. The Algebra of Bifunctions
39. Convex Processes .
Comments and References
Bibliography
Index
