# On the Behavior of Broyden's Class of Quasi-Newton Methods

Richard H. Byrd, Dong C. Liu, Jorge Nocedal

Research output: Contribution to journalArticlepeer-review

## Abstract

This paper analyzes algorithms from the Broyden class of quasi-Newton methods for nonlinear unconstrained optimization. This class depends on a parameter $\phi_k$, for which the choices $\phi_k = 0$ and $\phi_k = 1$ give the well-known BFGS and DFP methods. This paper examines algorithms that allow for negative values of the parameter $\phi_k$. It shows that severe restrictions have to be imposed on the selection of $\phi_k$ to guarantee q-superlinear convergence. It is argued that negative values of $\phi_k$ are desirable, and conditions on $\phi_k$ that guarantee superlinear convergence are given. However, practical algorithms that preserve the excellent properties of the BFGS method are not easy to design.
Original language English 533-557 SIAM Journal on Optimization 2 https://doi.org/10.1137/0802026 Published - 1992