Highly efficient broyden methods of minimization with variable parameter

Recently, there has been an increasing interest in the self-scaling Broyden family of formulae. These formulae are usually defined by replacing the approximate Hessian matrix B by τB for some scaling parameter τ. It is clear that if B is replaced by (1/τ)B in a self-scaling formula, then a member of the Broyden family follows. The author will show that in certain cases this member is not uniquely defined. He will illustrate this point by addressing new members of the Broyden family and giving a new approach to the self-dual update of Oren and Spedicato. The new members are defined by explicit expressions and satisfy the usual condition which ensures that the current Hessian approximations are maintained positive definite. Because comparison with the BFGS method shows promising numerical results, the corresponding new methods are efficient.