## 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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Given any set D C E, the linear span of D, denoted span (D), is the smallest linear subspace containing D. It consists exactly of all linear combinations of

**elements**of D. Analogously, the convex hull of D, denoted conv (D), ...

In fact any linear function from E to R has the form (a, -) for some

**element**a of E. Linear maps and affine functions (linear functions plus constants) are continuous. Thus, for example, closed halfspaces are indeed closed.

t t (c) Prove that for any set D C E, conv D is the set of all convex combinations of

**elements**of D. 3. Prove that a convex set D C E has convex closure, and deduce that cl (conv D) is the smallest closed convex set containing D. 4.

(b) Deduce that if in addition D and C are disjoint then there exists a nonzero

**element**a in E with infreD(a,a) > supyeo(a,y). Interpret geometrically. (c) Show part (b) fails for the closed convex sets in R*, D = {x ti > 0, x1a:2 > 1}, ...

(a) Prove the intersection of an arbitrary collection of affine sets is affine. (b) Prove that a set is affine if and only if it is a translate of a linear subspace. (c) Prove affD is the set of all affine combinations of

**elements**of D.

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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 |