## 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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The proof is outlined in Exercise 10.

**Exercises and Commentary**Good general references are [156] for elementary real analysis and [1] for linear algebra. Separation theorems for convex sets originate with Minkowski [129).

We leave the proof of this result as an exercise. ... We describe a proof of Fan's theorem in the exercises, using the above proposition and the following classical relationship between the set T" of doubly ...

**Exercises and Commentary**...

**Exercises and Commentary**Fan's inequality (1.2.2) appeared in [65], but is closely related to earlier work of von Neumann (163]. The condition for equality is due to [159]. The Hardy–Littlewood—Pólya inequality may be found in [82].

D Notice that the proof relies on consideration of a nondifferentiable function, even though the result concerns derivatives.

**Exercises and Commentary**The optimality conditions in this section are very standard (see for example [119).

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### Contents

15 | |

Fenchel Duality | 33 |

Convex Analysis | 65 |

Special Cases | 97 |

Nonsmooth Optimization | 123 |

KarushKuhnTucker Theory | 153 |

Fixed Points | 179 |

Infinite Versus Finite Dimensions | 209 |

List of Results and Notation | 221 |

Bibliography | 241 |

Index | 253 |

### Other editions - View all

Convex Analysis and Nonlinear Optimization: Theory and Examples Jonathan M. Borwein,Adrian S. Lewis No preview available - 2000 |