## 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 58

Just as for sets, geometric and topological ideas also intermingle for the functions

we study. Given a set D in E, we call a function f : D → R.

**continuous**(on D) if f(x')

— f(x) for any sequence c' – a in D. In this case it easy to check, for example, ...

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

will be crucial ...

Theorem 2.1.5 (Second order conditions) Suppose the twice

**continuously**

differentiable function f : R” – R has a critical point E. If E is a local minimizer then

the Hessian V*f(x) is positive semidefinite. Conversely, if the Hessian is positive ...

(The Rayleigh quotient) (a) Let the function f : R” \{0} – R be

**continuous**, satisfying

f(Ax) = f(a) for all X > 0 in R and nonzero a in R”. Prove f has a minimizer. (b)

Given a matrix A in S", define a function g(x) = x*Ax/||x||* for nonzero at in R”.

(iii) Deduce, furthermore, that any points y and z in E satisfy |Po(y) – Po(z)|| < |y –

z|, so in particular the projection Po : E → C is

**continuous**. (d) Given a nonzero

element a of E, calculate the nearest point in the subspace {a e E|(a,a) = 0} to the

...

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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 |