## 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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Our style is relatively informal; for example, the text of each section creates the context for many of the

**result**statements. We value the variety of independent, self-contained approaches over a single, unified, sequential development ...

218 10 List of

**Results**and Notation 221 10.1 Named

**Results**. . . . . . . . . . . . . . . . . . . . . . . . . 221 10.2 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 Bibliography 241 Index 253 Chapter 1 Background ...

The following

**result**, which we prove a little later, is both typical and fundamental. Theorem 1.1.1 (Basic separation) Suppose that the set C C E is closed and convex, and that the point y does not lie in C. Then there exist real b and ...

In this language the above

**result**states that the point y is separated from the set C by a hyperplane. In other words, C is contained in a certain closed halfspace whereas y is not. Thus there is a “dual” representation of C as the ...

The following

**result**is a prototype. The proof is a standard application of the Bolzano– Weierstrass theorem above. Proposition 1.1.3 (Weierstrass) Suppose that the set D C E is nonempty and closed, and that all the level sets of the ...

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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 |