Showing papers in "Ima Journal of Applied Mathematics in 1972"

Quasi-Newton Methods for Unconstrained Optimization

Abstract: Many techniques for solving general nonlinear unconstrained optimization problems involve iteratively minimizing a model function that satisfies certain interpolation conditions. These conditions provide a model that behaves like the objective function in the neighborhood of the current iterate. The model functions often involve second-order derivatives of the objective function, which can be expensive to calculate. The fundamental idea behind quasi-Newton methods is to maintain an approximation to the Hessian matrix. The practical success of quasi-Newton methods has spurred a great deal of interest and research that has resulted in a considerable number of variations of this idea. The analytical difficulties associated with characterizing the performance of these algorithms means there is a real need for practical testing to support theoretical claims. The goal of this project is to describe, implement, and test these methods in a way that is uniform, systematic, and consistent. In the first part of the paper, we derive several classical quasi-Newton methods, discuss their relative benefits, and show how to implement them. In the second part, we investigate more recent variations, explain their motivation and theory, and analyze their performance.

On Dual Extremum Principles in Applied Mathematics.

Abstract: : A unified account of upper and lower bounding principles involving convexity is given They bound an energy-type or cost functional Applications treated include linear and nonlinear programming, networks, optimization, control theory, fluid mechanics, elasticity, plasticity, and other differential, integral or operator equations and inequalities (Author)