## 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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A convex

**cone**which arises frequently in optimization is the

**normal**come to a convex set C at a point E € C, written No(a). This is the convex

**cone**of ...

In that case local minimizers a may not lie in the interior of the set C of interest, so the

**normal cone**NC(x) is not simply {0}. The next result shows that ...

Prove the

**normal cone**is a closed convex cone. 2. (Examples of

**normal cones**) For the following sets C C E, check C is convex and compute the

**normal cone**...

... prove the

**normal cone**to the set {a e E|Aa = b) at any point in it is A*Y. Hence deduce Corollary 2.1.3 (First order conditions for linear constraints).

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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 |