## 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 83

An easy exercise

**shows**that intersections of convex sets are convex. 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 ...

(Radstrom cancellation) Suppose sets A, B, C C E satisfy A + C C B + C. (a) If A and B are convex, B is closed, and C is bounded, prove A C B. (Hint: Observe 2A + C = A + (A + C) c 2B + C.) (b)

**Show**this result can fail if B is not ...

(c)

**Show**part (b) fails for the closed convex sets in R*, D = {x ti > 0, x1a:2 > 1}, C {x | x2 = 0}. 6. " (Recession cones) Consider a nonempty closed convex set C C E. We define the recession cone of C by 0+(C) = {d e E|C + R+d c C}.

This

**shows**, for example, that the function X is norm-preserving: |X| = |A(X)| for all X in S". For any X in S', the spectral decomposition also

**shows**there is a unique matrix X"/* in S' whose square is X. The Cauchy–Schwarz inequality ...

... is the function X e S' " log det X. An exercise

**shows**this function is differentiable on S' with derivative X". A convex cone which arises frequently in optimization is the normal cone to a convex set C at a point à e C, written No.

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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 |