## 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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Results 1-5 of 30

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), is the smallest ...

Given a

**subspace**G of E, the orthogonal complement of G is the

**subspace**G* = {

ye E|(t, y) = 0 for all a e G}, so called because we can write E as a direct ... Any

**subspace**G satisfies G** = G. The range of any linear map A coincides with N(A')"

.

(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 that 0

+(C) ...

(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. (d) Prove cl D C

affD and ...

(e) For any point a in D, prove affD = x+span (D–a), and deduce the linear

**subspace**span (D - a) is independent of a. 13. “ (The relative interior) (We use

Exercises 11 and 12.) The relative interior of a convex set C in E, denoted ri C, is

its ...

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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 |