## 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 37

166 7.4 Second

**Order Conditions**. . . . . . . . . . . . . . . . . . . 172 8 Fixed Points 179 8.1 The Brouwer Fixed Point Theorem . . . . . . . . . . . . . . 179 8.2 Selection and the Kakutani—Fan Fixed Point Theorem .

Chapter 2 Inequality Constraints 2.1 Optimality

**Conditions**Early in multivariate calculus we learn the ... a -ā) < 0 for all points a' in C. Proposition 2.1.1 (First

**order**necessary

**condition**) Suppose that C is a convex set in E and ...

This book is largely devoted to the study of first order necessary optimality conditions for a local minimizer of a ... The next result shows that when f is convex the first

**order condition**above is sufficient for à to be a global ...

In the absence of convexity, we need second order information to tell us more about minimizers. The following elementary result from multivariate calculus is typical. Theorem 2.1.5 (Second

**order conditions**) Suppose the twice ...

(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**...

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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 |