## 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 96

A set C in E is

**convex**if the line segment joining any two points a, ... and is

**closed**if D = cl D. Linear subspaces of E are important examples of

**closed**...

In other words, C is contained in a certain closed halfspace whereas y is not. ... of closed halfspaces, and is therefore both

**closed and convex**.

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 ...

Prove that a convex set D C E has convex closure, and deduce that cl (conv D) is the smallest

**closed convex**set containing D. 4.

(Strong separation) Suppose that the set C C E is

**closed and convex**, and that the set D C E is compact and convex. (a) Prove the set D – C is

**closed and**...

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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 |