## 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 6-10 of 55

The relative

**interior**of a convex set C in E, denoted riC, is its

**interior**relative to its affine hull. In other words, a point a lies in riC if there is a real 6 = 0 with (a + 6B) O affG C C. (a) Find convex sets C1 C C2 with riC1 Z ...

Prove S' is a closed convex cone with

**interior**S#1. 2. Explain why s: is not a polyhedron. 3. (S3 is not strictly convex) Find nonzero matrices X and Y in Sł such that R, X #RTY and (X +Y)/2 ZS:1. 4. (A nonlattice ordering) Suppose the ...

In that case local minimizers a may not lie in the

**interior**of the set C of interest, so the normal cone NC(x) is not simply {0}. The next result shows that when f is convex the first order condition above is sufficient for a to be a ...

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### Contents

1 | |

15 | |

Chapter 3 Fenchel Duality
| 33 |

Chapter 4 Convex Analysis
| 65 |

Chapter 5 Special Cases
| 97 |

Chapter 6 Nonsmooth Optimization
| 123 |

Chapter 7 KarushKuhnTucker Theory
| 153 |

Chapter 8 Fixed Points
| 179 |

Chapter 9 More N onsmooth Structure
| 213 |

Infinite Versus Finite Dimensions
| 239 |

Chapter 11 List of Results and Notation
| 253 |

Bibliography | 275 |

Index | 289 |

### Other editions - View all

Convex Analysis and Nonlinear Optimization: Theory and Examples Jonathan M. Borwein,Adrian S. Lewis No preview available - 2000 |