## 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 61

**Proposition**1.1.3 (Weierstrass) Suppose that the set D C E is nonempty and closed, and that all the level sets of the continuous function f : D → R are bounded. Then f has a global minimizer. Just as for sets, convexity of functions ...

**Proposition**1.1.5 For a convex set C C E, a convex function f : C – R. has bounded level sets if and only if it satisfies the growth condition (1.1.4). The proof is outlined in Exercise 10. Exercises and Commentary Good general ...

7. For any set of vectors a', a”, ..., a” in E, prove the function f(x) = max: (a', a) is convex on E. 8. Prove

**Proposition**1.1.3 (Weierstrass). 9. (Composing convex functions) Suppose that the set C C E is convex and that the functions ...

|am|| Hence complete the proof of

**Proposition**1.1.5. The relative interior Some arguments about finite-dimensional convex sets C simplify and lose no generality if we assume C contains 0 and spans E. The following exercises outline this ...

**Proposition**1.2.4 (Hardy–Littlewood—Pólya) Any vectors a and y in R” satisfy the inequality a"ys [x]"[y]. We describe a proof of Fan's theorem in the exercises, using the above

**proposition**and the following classical relationship ...

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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 |