## The Newton-Cauchy Framework: A Unified Approach to Unconstrained Nonlinear MinimizationComputational unconstrained nonlinear optimization comes to life from a study of the interplay between the metric-based (Cauchy) and model-based (Newton) points of view. The motivating problem is that of minimizing a convex quadratic function. This research monograph reveals for the first time the essential unity of the subject. It explores the relationships between the main methods, develops the Newton-Cauchy framework and points out its rich wealth of algorithmic implications and basic conceptual methods. The monograph also makes a valueable contribution to unifying the notation and terminology of the subject. It is addressed topractitioners, researchers, instructors, and students and provides a useful and refreshing new perspective on computational nonlinear optimization. |

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The Newton-Cauchy Framework: A Unified Approach to Unconstrained Nonlinear ... John L. Nazareth No preview available - 2006 |

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approach B-Update ball constraint basic Cauchy CG method CG-related Chapter choice computational linear algebra conceptual conjugate directions conjugate gradient conjugate gradient method consider convergence convex quadratic function corresponding gradient current iterate Davidon defined denote derived descent direction diagonal diagonal matrix direction dj direction of descent discussion eigenvalues equation equivalent Evaluate exact line search example finite follows given gj+i gradient change gradient vector Hessian approximation Hessian matrix implementation implies inverse Lagrange multiplier level-1 line search procedure linearly independent mathematical algorithm matrix H metric-based minimizing point Mjsj model-based Nazareth Newton's method nonsingular Note obtain orthogonal positive definite matrix problems Procedure QN/B Programming QN relation quadratic function quasi-Newton methods reconditioner result search direction vectors Section sequential exact line Sherman-Morrison formula solving SRI update step length strictly convex quadratic symmetric matrix tion transformation of variables variable metric Wolfe conditions