Convex Analysis

by Rockafellar, R. Tyrrell

• ISBN13: 9780691015866
• ISBN10: 0691015864
• eBook ISBN(s): 9781400873173
• Edition: Reprint
• Format: Paperback
• Copyright: 1996-12-23
• Publisher: Princeton Univ Pr

Summary

Available for the first time in paperback, R. Tyrrell Rockafellar's classic study presents readers with a coherent branch of nonlinear mathematical analysis that is especially suited to the study of optimization problems. Rockafellar's theory differs from classical analysis in that differentiability assumptions are replaced by convexity assumptions. The topics treated in this volume include: systems of inequalities, the minimum or maximum of a convex function over a convex set, Lagrange multipliers, minimax theorems and duality, as well as basic results about the structure of convex sets and the continuity and differentiability of convex functions and saddle- functions. This book has firmly established a new and vital area not only for pure mathematics but also for applications to economics and engineering. A sound knowledge of linear algebra and introductory real analysis should provide readers with sufficient background for this book. There is also a guide for the reader who may be using the book as an introduction, indicating which parts are essential and which may be skipped on a first reading.

Table of Contents

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