## 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 22

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.(T). This is the convex cone of normal

vectors, vectors d in E such that (d, a -ā) < 0 for all points a' in C. Proposition 2.1.1

...

In that case local minimizers á may not lie in the interior of the set C of interest, so

the

**normal cone**NC(#) is not simply {0}. The next result shows that when f is

convex the first order condition above is sufficient for à to be a global minimizer of

f ...

(Examples of

**normal cones**) For the following sets C C E, check C is convex and

compute the

**normal cone**No(£) for points à in C: (a) C a closed interval in R. (b)

C = B, the unit ball. (c) C a subspace. (d) C a closed halfspace: {a, (a,a) < b) ...

(Normals to affine sets) Given a linear map A : E → Y (where Y is a Euclidean

space) and a point b in Y, prove the

**normal cone**to the set {a e E Ax = b} at any

point in it is A*Y. Hence deduce Corollary 2.1.3 (First order conditions for linear ...

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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 |