## 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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88 5 Special Cases 97 5.1

**Polyhedral**Convex Sets and Functions . . . . . . . . . . . . 97 5.2 Functions of Eigenvalues . . . . . . . . . . . . . . . . . . . . 104 5.3 Duality for Linear and Semidefinite Programming .

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 (A"y, a) = (y, Aa) for all points a in E and y in ...

Xx > 0 for all vectors x in R”, and positive definite if the inequality is strict whenever a is nonzero.) These two cones have some important differences; in particular, R' is a

**polyhedron**, whereas the cone of positive semidefinite ...

Explain why s: is not a

**polyhedron**. (S: is not strictly convex) Find nonzero matrices X and Y in Sł such that RLX #REY and (X +Y)/2 g S$4. (A nonlattice ordering) Suppose the matrix Z in S satisfies 0 0 0 1 1 0 w=|| || |and w=| | <= W ...

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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 |