A double parameter scaled BFGS method for unconstrained optimization

A double parameter scaled BFGS method for unconstrained optimization is presented. In this method, the first two terms of the known BFGS update formula are scaled with a positive parameter while the third one is scaled with another positive parameter. These parameters are selected in such a way as to improve the eigenvalues structure of the BFGS update. The parameter scaling the first two terms of the BFGS update is determined by clustering the eigenvalues of the scaled BFGS matrix. On the other hand, the parameter scaling the third term is determined as a preconditioner to the Hessian of the minimizing function combined with the minimization of the conjugacy condition from conjugate gradient methods. Under the inexact Wolfe line search, the global convergence of the double parameter scaled BFGS method is proved in very general conditions without assuming the convexity of the minimizing function. Using 80 unconstrained optimization test functions with a medium number of variables, the preliminary numerical experiments show that this double parameter scaled BFGS method is more efficient than the standard BFGS update or than some other scaled BFGS methods.