## Convex Analysis and Nonlinear Optimization: Theory and ExamplesOptimization is a rich and thriving mathematical discipline. The theory underlying current computational optimization techniques grows ever more sophisticated. The powerful and elegant language of convex analysis unifies much of this theory. The aim of this book is to provide a concise, accessible account of convex analysis and its applications and extensions, for a broad audience. It can serve as a teaching text, at roughly the level of first year graduate students. While the main body of the text is self-contained, each section concludes with an often extensive set of optional exercises. The new edition adds material on semismooth optimization, as well as several new proofs that will make this book even more self-contained. |

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Results 1-5 of 89

7.2 The Karush–Kuhn–Tucker

**Theorem**. . . . . . . . . . . . . 7.3 Metric Regularity and the Limiting Subdifferential . . . . . 7.4 Second Order Conditions .

**Theorem**1.2.1 (Fan) Any matrices X and Y in S" satisfy the inequality tr (XY) < A(X)"X(Y). (1.2.2) Equality holds if and only if X and Y have a simultaneous ...

2.2

**Theorems**of the Alternative One well-trodden route to the study of ... Our first approach, however, relies on a different

**theorem**of the alternative.

D > 0 The equivalence of (ii) and (iii) gives Gordan's

**theorem**. We now proceed by using Gordan's

**theorem**to derive the Farkas lemma, one of the cornerstones ...

D Just like Gordan's

**theorem**, the Farkas lemma has an important geometric interpretation which gives an alternative approach to its proof (Exercise 6): any ...

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### Contents

1 | |

15 | |

Chapter 3 Fenchel Duality
| 33 |

Chapter 4 Convex Analysis
| 65 |

Chapter 5 Special Cases
| 97 |

Chapter 6 Nonsmooth Optimization
| 123 |

Chapter 7 KarushKuhnTucker Theory
| 153 |

Chapter 8 Fixed Points
| 179 |

Chapter 9 More N onsmooth Structure
| 213 |

Infinite Versus Finite Dimensions
| 239 |

Chapter 11 List of Results and Notation
| 253 |

Bibliography | 275 |

Index | 289 |

### Other editions - View all

Convex Analysis and Nonlinear Optimization: Theory and Examples Jonathan M. Borwein,Adrian S. Lewis No preview available - 2000 |