Enforcing convergence to all members of the Broyden family of methods for unconstrained optimization

The Broyden family of quasi-Newton methods for unconstrained optimization will be considered. It is well-known that if a member of this family is defined su ciently close to that of the robust BFGS method, then useful theoretical and numerical properties are obtained. These properties will be extended to all members of the above family provided that the current points are su ciently close to the solution of a convex optimization problem and that the Hessian approximations satisfy certain conditions. A possibility for enforcing these properties to all members of the family will be provided for any starting point and any initial positive definite Hessian approximation. Numerical results will be described to illustrate that some robust, ine cient and divergent Broyden family methods are enforced to be competitive with the standard BFGS method.