## 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. |

### From inside the book

Results 1-5 of 41

**Corollary**2.1.3 (First order conditions for linear constraints) For a convex set C c

E, a function f : C – R, a linear map A: E → Y (where Y is a Euclidean space) and

a point b in Y, consider the optimization problem inf{f(x)|x e C, Ax = b}. (2.1.4) ...

To illustrate the effect of constraints on second order conditions, consider the

framework of

**Corollary**2.1.3 (First order conditions for linear constraints) in the

case E = R”, and suppose V f(x) e A*Y and f is twice continuously differentiable

near 3 ...

(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 ...

By considering the problem (for C eS'') inf{(C, X) – log det X |X e Ln S''), use

Section 1.2, Exercise 14 and

**Corollary**2.1.3 (First order conditions for linear

constraints) to prove there exists a matrix X in £S' with C – X" having (i,j)th entry of

zero ...

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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 |