## 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 35

A

**set**C in E is

**convex**if the line segment joining any two points a and y in C is contained in C: algebraically, ... of all

**convex**combinations of elements of D, that is to say points of the form XX"I Aiz', where A, e R+ and

**c e**D for ...

Just as for sets, convexity of functions will be crucial for us. Given a

**convex set C C E**, we say that the function f : C → R is convex if f(\a + (1 – A)y) < \f(x) + (1 - A)f(y) for all points a, and y in C and 0 < X s. 1.

Proposition 1.1.5 For a

**convex set C C E**, a convex function f : C – R. has bounded level sets if and only if it satisfies the growth condition (1.1.4). The proof is outlined in Exercise 10. Exercises and Commentary Good general ...

(Strong separation) Suppose that the

**set C C E**is closed and

**convex**, and that the

**set**D C E is compact and

**convex**. (a) Prove the

**set**D – C is closed and

**convex**. (b) Deduce that if in addition D and C are disjoint then there exists a ...

D The case of this result where C is an open

**set**is the canonical introduction to the use of calculus in ... 2.1.2 (First order sufficient condition) Suppose that the

**set C C E**is

**convex**and that the

**function**f : C → R is

**convex**.

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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 |