## 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 72

If C is

**nonempty**and satisfies R+C = C we call it a come. (Notice we require that cones contain the origin.) Examples are the positive orthant R* = {a e R” ...

Proposition 1.1.3 (Weierstrass) Suppose that the set D C E is

**nonempty**and closed, and that all the level sets of the continuous function f : D → R are ...

(Recession cones) Consider a

**nonempty**closed convex set C C E. We define the recession cone of C by 0" (C) = {d e E|C + Rid C C).

(e) Deduce further that if int C is

**nonempty**then cl (int C) = clC. Is convexity necessary? 12. " (Affine sets) A set L in E is affine if the entire line ...

w"} of E. Deduce (1/(n + 1)) XD'a e int C. Hence deduce that any

**nonempty**convex set in E has

**nonempty**relative interior. (c) Prove that for 0 < \s 1 we ...

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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 |