The most successful quasi-Newton methods for solving unconstrained optimization problems when second derivatives are unavailable or expensive have used the BFGS update. In recent tests, Brodlie reports that an update introduced by Biggs performs equally well. This update differs from the BFGS in that it alters the secant equation to incorporate information from a cubic model. In this paper we show that Biggs’ method retains the Q-superlinear convergence properties of the BFGS exhibited by Broyden, Denis, and More and by Powell. Our proofs show that near the solution of most problems, Biggs’ method and the BFGS are essentially the same. We also establish necessary and sufficient conditions for the Q-superlinear convergence of a general class of quasi-Newton methods which modify the secant equation similarly to Biggs.
Schnabel, Robert B., "Q-Superlinear Convergence of Biggs' Method and Related Methods for Unconstrained Optimization ; CU-CS-133-78" (1978). Computer Science Technical Reports. 131.