## Convex Analysis and Monotone Operator Theory in Hilbert SpacesThis book provides a largely self-contained account of the main results of convex analysis and optimization in Hilbert space. A concise exposition of related constructive fixed point theory is presented, that allows for a wide range of algorithms to construct solutions to problems in optimization, equilibrium theory, monotone inclusions, variational inequalities, best approximation theory, and convex feasibility. The book is accessible to a broad audience, and reaches out in particular to applied scientists and engineers, to whom these tools have become indispensable. |

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### Contents

Background
| 1 |

Hilbert Spaces
| 27 |

Convex Sets
| 43 |

Convexity and Nonexpansiveness
| 59 |

Fejér Monotonicity and Fixed Point Iterations
| 75 |

Convex Cones and Generalized Interiors
| 87 |

Support Functions and Polar Sets
| 107 |

Convex Functions
| 113 |

Further Differentiability Results
| 261 |

Duality in Convex Optimization
| 275 |

Monotone Operators
| 293 |

Finer Properties of Monotone Operators
| 310 |

Stronger Notions of Monotonicity
| 323 |

Resolvents of Monotone Operators
| 332 |

Sums of Monotone Operators
| 351 |

Zeros of Sums of Monotone Operators
| 363 |

Lower Semicontinuous Convex Functions
| 128 |

Convex Functions Variants
| 143 |

Convex Variational Problems
| 154 |

Infimal Convolution
| 167 |

Conjugation
| 181 |

Further Conjugation Results
| 196 |

FenchelRockafellar Duality
| 207 |

Subdifferentiability
| 223 |

Differentiability of Convex Functions
| 241 |

Fermats Rule in Convex Optimization
| 381 |

Proximal Minimization
| 398 |

Projection Operators
| 415 |

Best Approximation Algorithms
| 431 |

Bibliographical Pointers | 441 |

Symbols and Notation | 443 |

449 | |

461 | |

### Other editions - View all

Convex Analysis and Monotone Operator Theory in Hilbert Spaces Heinz H. Bauschke,Patrick L. Combettes No preview available - 2013 |

Convex Analysis and Monotone Operator Theory in Hilbert Spaces Heinz H. Bauschke,Patrick L. Combettes No preview available - 2011 |

### Common terms and phrases

a e H algorithm Argmin f closed convex subset cluster point converges weakly Convex Analysis convex cone convex function convex set deduce epif Example Exercise exists f and g f is convex Fejér firmly nonexpansive following hold follows from Corollary follows from Proposition Fréchet differentiable function f functions in To(H Gâteaux differentiable H and let H is finite-dimensional H x H Hausdorff space Hence implies inf f(H int dom f Lemma let a e let a e H let an)nen Let f e To(H let Le B(H linear subspace Lipschitz continuous lower semicontinuous monotone operator nonempty closed convex nonempty subset nonexpansive operators oo be proper Proof real Hilbert space sequential cluster point Set f Show Springer Science+Business Media strictly convex subset of H Suppose that H u e H weakly sequentially yields