## 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 33

**Calculate**the adjoint map A”. * (Fan's inequality) For vectors x and y in R” and a matrix U in O”, define a = (Diaga, U"(Diagy)0). (a) Prove a = x*Zy for some doubly stochastic matrix Z. (b) Use Birkhoff's theorem and Proposition 1.2.4 ...

(c)

**Calculate**Vg(x) for nonzero a. (d) Deduce that minimizers of g must be eigenvectors, and

**calculate**the minimum value. (e) Find an alternative proof of part (d) by using a spectral decomposition of A. (Another approach to this ...

(d) Given a nonzero element a of E,

**calculate**the nearest point in the subspace {a e E|(a,a) = 0} to the point ye E. (e) (Projection on R' and S') Prove the nearest point in R' to a vectory in R” is y", where y' = max{yi,0} for each i.

... space of complex polynomials of degree no more than n, with inner product 71. - 7. • 7% (XX,” ? XXuz') : XXiu. j=0 j=0 j=0 Given a polynomial p in this space,

**calculate**the nearest 2.1 Optimality Conditions 21.

Given a polynomial p in this space,

**calculate**the nearest polynomial with a given complex root a, and prove the distance to this polynomial is (XX_o|al”)"/"|p(a). 2.2 Theorems of the Alternative One well-trodden route to the 22 2.

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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 |