## Differentiable Optimization and Equation Solving: A Treatise on Algorithmic Science and the Karmarkar RevolutionIn 1984, N. Karmarkar published a seminal paper on algorithmic linear programming. During the subsequent decade, it stimulated a huge outpouring of new algorithmic results by researchers world-wide in many areas of mathematical programming and numerical computation. This book gives an overview of the resulting, dramatic reorganization that has occurred in one of these areas: algorithmic differentiable optimization and equation-solving, or, more simply, algorithmic differentiable programming. The book is aimed at readers familiar with advanced calculus, numerical analysis, in particular numerical linear algebra, the theory and algorithms of linear and nonlinear programming, and the fundamentals of computer science, in particular, computer programming and the basic models of computation and complexity theory. J.L. Nazareth is a Professor in the Department of Pure and Applied Mathematics at Washington State University. He is the author of two books previously published by Springer-Verlag, DLP and Extensions: An Optimization Model and Decision Support System (2001) and The Newton-Cauchy Framework: A Unified Approach to Unconstrained Nonlinear Minimization (1994). |

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### Contents

1 | 3 |

Algorithmic Science | 9 |

2 | 11 |

EulerNewton and LagrangeNC Methods | 17 |

2 1 Newton 2 2 2 QuasiNewton 2 2 3 Limited Memory | 22 |

Introduction The EN Method | 28 |

The LNC Method | 35 |

Notes | 45 |

LogBarrier Transformations | 146 |

Karmarkar Potentials and Algorithms | 155 |

Algorithmic Principles | 166 |

A New Paradigm | 179 |

1 1 Notation 12 1 2 Unweighted Case 12 1 3 Weighted Case A PotentialReduction Algorithm | 186 |

3 | 187 |

6 | 193 |

An Emerging Discipline | 204 |

### Other editions - View all

Differentiable Optimization and Equation Solving: A Treatise on Algorithmic ... John Nazareth No preview available - 2003 |

Differentiable Optimization and Equation Solving: A Treatise on Algorithmic ... John Nazareth No preview available - 2013 |

### Common terms and phrases

affine scaling affine-scaling algorithm algorithms based approach associated BFGS BFGS update Cauchy central path Chapter choice components computer science conceptual constraints convex corresponding current iterate defined denotes diagonal matrix direction of descent directional derivative discussion Dk+1 equation-solving equations Euler predictor evolutionary evolutionary algorithms example f-values feasible interior follows foregoing formulation G-nome G-type given gradient vector Hessian Hessian matrix homotopy path homotopy system 10.2 implementation initial interior path interior-point Jacobian matrix Karmarkar’s L-BFGS line search linear programming major cycle Mathematical Programming merit function method Mk+1 multialgorithm Nazareth Nelder–Mead Newton corrector Newton–Cauchy NM-GS objective function obtained optimization orthant parameter particular path-following population positive definite potential function primal and dual quadratic quantities quasi-Newton real-number restart rithms routine search direction Section simplex algorithm solution solving starting point step strategy techniques tion transformations trust region Turing machine variable-metric variables xk+1