## 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 94

Where possible, we have chosen a

**proof**technique permitting those readers

familiar with functional analysis to discover for themselves how a result extends.

We would, in part, like this book to be an entrée for mathematicians to a valuable

...

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 continuous function f : D → R are

bounded.

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). The theory

of the ...

|am|| Hence complete the

**proof**of Proposition 1.1.5. The relative interior Some

arguments about finite-dimensional convex sets C simplify and lose no generality

if we assume C contains 0 and spans E. The following exercises outline this idea.

We leave the

**proof**of this result as an exercise. Proposition 1.2.4 (Hardy–

Littlewood—Pólya) Any vectors a and y in R” satisfy the inequality a"ys [x]"[y]. We

describe a

**proof**of Fan's theorem in the exercises, using the above proposition

and 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 |