## 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 88

**Deduce**that the convex hull of a set D C E is well-defined as the intersection of

all convex sets containing D. 2. (a) Prove that if the set C C E is convex and if a", r

",..., x" e C, 0 < A1, X2,..., Xn e R, and XXi = 1 then XXia e C. Prove, furthermore, ...

(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}, C {x | x2 = 0}

. 6.

**Deduce**the existence of a sequence (a.”) in C with f(x”) < |x"|/m – +oo. For a fixed

point à in C, derive a contradiction by considering the sequence 7m, à -- T.T(x" –

3). |am|| Hence complete the proof of Proposition 1.1.5. The relative interior ...

(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 ...

(a) Prove a = x*Zy for some doubly stochastic matrix Z. (b) Use Birkhoff's theorem

and Proposition 1.2.4 to

**deduce**the inequality a < [a]"[y]. (c)

**Deduce**Fan's

inequality (1.2.2). (A lower bound) Use Fan's inequality (1.2.2) for two matrices X

and ...

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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 |