## 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 54

After a new

**sequence**of exercises ending Chapter 8 with a concise approach to monotone operator theory via convex analysis, the new Chapter 9 begins with a ...

We say the point x in E is the limit of the

**sequence**of points a ', a. ... The closure of D is the set of limits of

**sequences**of points in D, written cl D, ...

Theorem 1.1.2 (Bolzano–Weierstrass) Bounded

**sequences**in E have ... (on D) if f(x') — f(x) for any

**sequence**a' = a in D. In this case it easy to check, ...

(d) For a unit vector u in E, prove u e 0" (C) if and only if there is a

**sequence**(a") in C satisfying ||a"| – Co and |x"| "a" – u.

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), ...

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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 |