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On Recent Developments in BFGS Methods for Unconstrained Optimization

by Philip E. Gill, Jeb H. Runnoe


Quasi-Newton methods form the basis of many effective methods for unconstrained and constrained optimization. As the iterations proceed, a quasi-Newton method incorporates new curvature information by performing a low-rank update to a matrix that serves as an approximation to a Hessian matrix of second derivatives. In the years following the publication of the Davidon-Fletcher-Powell (DFP) method in 1963 the Broyden-Fletcher-Goldfarb-Shanno (BFGS) update emerged as the best update formula for use in unconstrained minimization. More recently, a number of quasi-Newton methods have been proposed that are intended to improve on the efficiency and reliability of the BFGS method. Unfortunately, there is no known analytical means of determining the relative performance of these methods on a general nonlinear function, and there is a real need for extensive experimental testing to justify the theoretical basis of each approach. The goal of this report is to implement and test these methods in a uniform, systematic, and consistent way. In the first part of the report, we review several quasi-Newton methods, discuss their relative benefits, and discuss their implementation. In the second part, we investigate more recent variations, explain their motivation and theory, and investigate their performance.

UCSD-CCoM-22-04.pdf   July 2022