## 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 43

(Accessibility

**lemma**) Suppose C is a convex set in E. (a) Prove clo' C. C + eB for any real e > 0. (b) For sets D and F in E with D open, ...

(Singular values and von Neumann's

**lemma**) Let M" denote the vector space of n x n real matrices. For a matrix A in M” we define the singular values of A by ...

(d) By considering matrices of the form A+ al and B + 61, deduce Fan's inequality from von Neumann's

**lemma**(part (b)). Chapter 2 Inequality Constraints 2.1 ...

... the Farkas

**lemma**(which we derive at the end of this section). Our first approach, however, relies on a different theorem of the alternative. 1.

We now proceed by using Gordan's theorem to derive the Farkas

**lemma**, one of the cornerstones of many approaches to optimality conditions.

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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 |