Convex Functions (eBook)

Front Cover 1
Convex Functions 4
Copyright Page 5
Contents 8
Preface 12
Acknowledgments 14
Introduction 16
List of Notation 20
Chapter I. Convex Functions on the Real Line 22
10. Introduction 23
11. Continuity and Differentiability 24
12. Characterizations 30
13. Closure under Functional Operations 36
14. Differences of Convex Functions 43
15. Conjugate Convex Functions 49
Chapter II. Normed Linear Spaces 58
20. Introduction 59
21. Normed Linear Spaces 59
22. Functions on Normed Linear Spaces 75
23. Derivatives in a Normed Linear Space 83
Chapter III. Convex Sets 93
30. Introduction 94
31. Convex Sets and Affine Sets 94
32. Hyperplanes and Extreme Points 102
Chapter IV. Convex Functions on a Normed Linear Space 109
40. Introduction 110
41. Continuity of Convex Functions 112
42. Differentiable Convex Functions 118
43. The Support of Convex Functions 125
44. Differentiability of Convex Functions 134
Chapter V. Optimization 142
50. Introduction 143
51. Maxima and Minima 143
52. Minimax Theorems and the Theory of Games 149
53. Linear Programming 159
54. The Simplex Method 175
55. Convex Programming 191
56. Approximation 200
Chapter VI. Inequalities 209
60. Introduction 210
61. The Classical Inequalities 210
62. The Generalized Geometric Mean–Arithmetic Mean Inequality and Norms 215
63. Matrix Inequalities 221
Chapter VII. Midconvex Functions 231
70. Introduction 232
71. Midconvex Functions on a Normed Linear Space 232
72. Midconvex Functions on R 239
Chapter VIII. Related Classes of Functions 247
80. Introduction 248
81. Quasiconvex Functions 249
82. Completely Convex Functions 254
83. Convex Functions of Higher Order 258
84. Generalized Convex Functions 261
85. More about Generalized Convex Functions 267
86. Other Related Topics 274
Appendix 284
Introduction 285
Independent Study Projects 285
Unsolved Problems 292
Bibliography 294
Author Index 310
Subject Index 316