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.
| Preface | ||
| Introductory Remarks: A Guide for the Reader | ||
| 1 | Affine Sets | 3 |
| 2 | Convex Sets and Cones | 10 |
| 3 | The Algebra of Convex Sets | 16 |
| 4 | Convex Functions | 23 |
| 5 | Functional Operations | 32 |
| 6 | Relative Interiors of Convex Sets | 43 |
| 7 | Closures of Convex Functions | 51 |
| 8 | Recession Cones and Unboundedness | 60 |
| 9 | Some Closedness Criteria | 72 |
| 10 | Continuity of Convex Functions | 82 |
| 11 | Separation Theorems | 95 |
| 12 | Conjugates of Convex Functions | 102 |
| 13 | Support Functions | 112 |
| 14 | Polars of Convex Sets | 121 |
| 15 | Polars of Convex Functions | 128 |
| 16 | Dual Operations | 140 |
| 17 | Caratheodory's Theorem | 153 |
| 18 | Extreme Points and Faces of Convex Sets | 162 |
| 19 | Polyhedral Convex Sets and Functions | 170 |
| 20 | Some Applications of Polyhedral Convexity | 179 |
| 21 | Helly's Theorem and Systems of Inequalities | 185 |
| 22 | Linear Inequalities | 198 |
| 23 | Directional Derivatives and Subgradients | 213 |
| 24 | Differential Continuity and Monotonicity | 227 |
| 25 | Differentiability of Convex Functions | 241 |
| 26 | The Legendre Transformation | 251 |
| 27 | The Minimum of a Convex Function | 263 |
| 28 | Ordinary Convex Programs and Lagrange Multipliers | 273 |
| 29 | Bifunctions and Generalized Convex Programs | 291 |
| 30 | Adjoint Bifunctions and Dual Programs | 307 |
| 31 | Fenchel's Duality Theorem | 327 |
| 32 | The Maximum of a Convex Function | 342 |
| 33 | Saddle-Functions | 349 |
| 34 | Closures and Equivalence Classes | 359 |
| 35 | Continuity and Differentiability of Saddle-functions | 370 |
| 36 | Minimax Problems | 379 |
| 37 | Conjugate Saddle-functions and Minimax Theorems | 388 |
| 38 | The Algebra of Bifunctions | 401 |
| 39 | Convex Processes | 413 |
| Comments and References | 425 | |
| Bibliography | 433 | |
| Index | 447 |
Overview
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 ...