## 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 6-10 of 42

In the absence of convexity, we need second order information to tell us more

about

**minimizers**. ... (In fact for T to be a local

**minimizer**it is sufficient for the

Hessian to be positive semidefinite locally, the function a € R → a highlights the ...

The function f + e|| || has bounded level sets, so has a global

**minimizer**a by the

Weierstrass proposition (1.1.3). If the vector d = Vf(x*) satisfies ||d|| > 6 then, from

the inequality +--vie),4)--a <-la. we would have for small t > 0 the contradiction ...

Prove that the differentiable function x* + x;(1 – a 1)" has a unique critical point in

R”, which is a local

**minimizer**, but has no global

**minimizer**. Can this happen on

R2 6. (The Rayleigh quotient) (a) Let the function f : R” \{0} – R be continuous, ...

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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 |