## 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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This is for reasons of space and accessibility rather than history or application: convex analysis developed ... BORWEIN ADRIAN S. LEWIS Gargnano, Italy September 1999 Contents Preface Vii 1 Background 1 1.1

**Euclidean Spaces**. viii Preface.

Chapter 1 Background 1.1

**Euclidean Spaces**We begin by reviewing some of the fundamental algebraic, geometric and analytic ideas we use throughout the book. Our setting, for most of the book, is an arbitrary

**Euclidean space**E, ...

... f(x) for any sequence c' – a in D. In this case it easy to check, for example, that for any real a the level set {x e D |f(x) < a.} is closed providing D is closed. Given another

**Euclidean space**Y, we call a map A : E → Y linear ...

(e) If Y is a

**Euclidean space**, the map A : E → Y is linear, and N(A)n 0+(C) is a linear subspace, prove AC is closed. Show this result can fail without the last assumption. (f) Consider another nonempty closed convex set D C E such ...

... e > 0 with a +é(a:-y) in C. (iii) R+(C - a) is a linear subspace. (e) If F is another

**Euclidean space**and the map A : E – F is linear, prove riAC D Ari C. 1.2 Symmetric Matrices Throughout most of this book our setting 1. Background.

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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 |