## Numerical Optimization: theoretical and practical aspects : with 26 figuresStarting with illustrative real-world examples, this book exposes in a tutorial way algorithms for numerical optimization: fundamental ones (Newtonian methods, line-searches, trust-region, sequential quadratic programming, etc.), as well as more specialized and advanced ones (nonsmooth optimization, decomposition techniques, and interior-point methods). Most of these algorithms are explained in a detailed manner, allowing straightforward implementation. Theoretical aspects are addressed with care, often using minimal assumptions. The present version contains substantial changes with respect to the first edition. Part I on unconstrained optimization has been completed with a section on quadratic programming. Part II on nonsmooth optimization has been thoroughly reorganized and expanded. In addition, nontrivial application problems have been inserted, in the form of computational exercises. These should help the reader to get a better understanding of optimization methods beyond their abstract description, by addressing important features to be taken into account when passing to implementation of any numerical algorithm. This level of detail is intended to familiarize the reader with some of the crucial questions of numerical optimization: how algorithms operate, why they converge, difficulties that may be encountered and their possible remedies. |

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### Contents

General Introduction | 3 |

Basic Methods | 25 |

LineSearches | 37 |

Newtonian Methods | 51 |

Conjugate Gradient | 67 |

Special Methods | 77 |

Some Theory of Nonsmooth Optimization | 95 |

Some Methods in Nonsmooth Optimization | 101 |

Globalization by LineSearch | 235 |

QuasiNewton Versions | 265 |

InteriorPoint Algorithms for Linear and Quadratic | 283 |

Linearly Constrained Optimization and Simplex Algorithm291 | 291 |

Linear Monotone Complementarity and Associated | 307 |

PredictorCorrector Algorithms | 329 |

NonFeasible Algorithms | 345 |

SelfDuality | 357 |

Bundle Methods The Quest of Descent | 117 |

Decomposition and Duality | 137 |

Newtons Methods in Constrained Optimization | 149 |

Background | 157 |

Local Methods for Problems with Equality Constraints | 169 |

Local Methods for Problems with Equality and Inequality | 203 |

Exact Penalization | 217 |

OneStep Methods | 367 |

Complexity of Linear Optimization Problems with | 383 |

Karmarkars Algorithm | 389 |

397 | |

415 | |

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### Common terms and phrases

affine function approximate assumption asymptotic augmented Lagrangian BFGS bounded bundle methods central path Chap components compute conjugate gradient Consider convex convex function cutting-planes cutting-planes method decomposition decrease deduce defined denote descent direction differentiable direction dk dual problem equation example exists f(xk feasible set finite formula global convergence hence holds implies iteration Lagrange multiplier Lemma line-search linear complementarity problem linear system Lipschitz continuous matrix merit function minimizing minimum multiplier neighborhood Newton Newton's method nonlinear nonsingular norm notation null space objective function obtain optimality conditions parameter positive definite primal primal-dual solution Proof proposition quadratic convergence quasi-Newton quasi-Newton methods reduced Hessian result right inverse S(LCP satisfies second-order sequence xk simplex algorithm solve SQP algorithm stationary point stepsize strict complementarity subgradient subproblems Suppose surjective tangent quadratic problem Theorem update variables vector xk+1 xk+i yk+1

### Popular passages

Page 398 - RH Byrd. ME Hribar, and J. Nocedal. An interior point algorithm for large scale nonlinear programming.

Page 402 - PE Gill, W. Murray, and MA Saunders, SNOPT: An SQP algorithm for large-scale constrained optimization, Numerical Analysis Report 97-2, Department of Mathematics, University of California, San Diego, La Jolla, CA, (1997).