## 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. |

### From inside the book

Results 1-5 of 95

**Deduce**that the convex hull of a set D C E is well-defined as the intersection of all convex sets containing D. 2. (a) (c) Prove that if the set C C E is ...

(b)

**Deduce**that if in addition D and C are disjoint then there exists a nonzero element a in E with infred (a,a) > supyeo(a, y). Interpret geometrically.

**Deduce**the existence of a sequence (a") in C with f(a.”) < |x"|/m – +oo. For a fixed point à in C, derive a contradiction by considering the sequence 'n), ...

(e) For any point a in D, prove affD = a +span (D–a), and

**deduce**the linear subspace span (D - a) is independent of a. 13. “ (The relative interior) (We use ...

**Deduce**the inequality 77, 'S - (II*)" 1. ... stochastic matrix Z. (b) Use Birkhoff's theorem and Proposition 1.2.4 to

**deduce**the inequality a < [a]"[y].

### What people are saying - Write a review

### 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 |