Conditioning of Quasi-Newton Methods for Function Minimization
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TLDR
In this paper, a class of approximating matrices as a function of a scalar parameter is presented, where the problem of optimal conditioning of these matrices under an appropriate norm is investigated and a set of computational results verifies the superiority of the new methods arising from conditioning considerations to known methods.Abstract:
Quasi-Newton methods accelerate the steepest-descent technique for function minimization by using computational history to generate a sequence of approximations to the inverse of the Hessian matrix. This paper presents a class of approximating matrices as a function of a scalar parameter. The problem of optimal conditioning of these matrices under an appropriate norm as a function of the scalar parameter is investigated. A set of computational results verifies the superiority of the new methods arising from conditioning considerations to known methods.read more
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References
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Journal ArticleDOI
A Rapidly Convergent Descent Method for Minimization
Roger Fletcher,M. J. D. Powell +1 more
TL;DR: A number of theorems are proved to show that it always converges and that it converges rapidly, and this method has been used to solve a system of one hundred non-linear simultaneous equations.
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A family of variable-metric methods derived by variational means
TL;DR: In this paper, a rank-two variable-metric method was derived using Greenstadt's variational approach, which preserves the positive-definiteness of the approximating matrix.
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A Class of Methods for Solving Nonlinear Simultaneous Equations
TL;DR: In this article, the authors discuss certain modifications to Newton's method designed to reduce the number of function evaluations required during the iterative solution process of an iterative problem solving problem, such that the most efficient process will be that which requires the smallest number of functions evaluations.
Journal ArticleDOI
Quasi-Newton methods and their application to function minimisation
TL;DR: The Newton-Raphson method as mentioned in this paper is one of the most commonly used methods for solving nonlinear problems, where the corrections are computed as linear combinations of the residuals.
Journal ArticleDOI
A Comparison of Several Current Optimization Methods, and the use of Transformations in Constrained Problems
TL;DR: Transitions whereby inequality constraints of certain forms can be eliminated from the formulation of an optimization problem are described, and examples of their use compared with other methods for handling such constraints are described.