## 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 31

Just as for

**sets**, geometric and topological ideas also intermingle for the ... any real a the

**level**set {a € D |f(a) < 0 } is closed providing D is closed.

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 ...

Proposition 1.1.5 For a convex set C C E, a convex function f : C → R. has bounded

**level sets**if and only if it satisfies the growth condition (1.1.4).

10 * (Convex growth conditions) (a) Find a function with bounded

**level sets**which does not satisfy the growth condition (1.1.4). (b) Prove that any function ...

(A lower bound) Use Fan's inequality (1.2.2) for two matrices X and Y in S" to prove a lower bound for tr(XY) in terms of A(X) and A(Y). 14. " (

**Level sets**...

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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 |