## 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 80

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, Ax + (1 ... Theorem 1.1.1 (Basic separation)

Suppose that the set C C E is

**closed and convex**, and that the point y does not lie

in C.

Thus, for example, closed halfspaces are indeed closed. A polyhedron is a finite

intersection of closed halfspaces, and is therefore both

**closed and convex**. The

adjoint of the map A above is the linear map A* : Y → E defined by the property ...

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 ...

t t (c) Prove that for any set D C E, conv D is the set of all convex combinations of

elements of D. 3. Prove that a convex set D C E has convex closure, and deduce

that cl (conv D) is the smallest

**closed convex**set containing D. 4. (Radstrom ...

(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

nonzero ...

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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 |