## 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 32

(Radstrom cancellation) Suppose sets A, B, C C E satisfy A + C C B + C. (a) If A

and B are convex, B is closed, and C is bounded, prove A C B. (

**Hint**: Observe 2A

+ C = A + (A + C) c 2B + C.) (b) Show this result can fail if B is not convex. 5.

(

**Hint**: Consider the relationship (X1/2 + Y1/2)a, (X1/2 – Y1/2)a) = (X - Y)", r) = 0,

for eigenvectors x of X*/* – Y"/2.) " (Square-root iteration) Suppose a matrix A in S'

satisfies I - A. Prove that the iteration 1 Y0 = 0, Yari =#(A+Y#) (n = 0,1,2,.

(

**Hint**: Use Exercise 13.) 15. " (Theobald's condition) Assuming Fan's inequality (

1.2.2), complete the proof of Fan's theorem (1.2.1) as follows. Suppose equality

holds in Fan's inequality (1.2.2), and choose a spectral decomposition X + Y ...

(Coercivity) Suppose that the function f : E → R is differentiable and satisfies the

growth condition limir .22 f(x)/|x|= +co. Prove that the gradient map Vf has range E

. (

**Hint**: Minimize the function f() — (a, -) for elements a of E.) 10. (a) Prove 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 |