Properties and numerical performance of quasi-Newton methods with modified quasi-Newton equations

Quasi-Newton (QN) equation plays a core role in contemporary nonlinear optimization. The usual QN equation employs only the gradients, but ignores the available function value information. In this paper, we derive a class of modified QN equations with a vector parameter which use both available gradient and function value information. The modified quasi-Newton methods maintain most properties of the usual quasi-Newton methods, meanwhile they achieve a higher-order accuracy in approximating the second-order curvature of the problem functions than the usual ones do. Numerical experiments are reported which support the theoretical analyses and show the advantages of the modified QN methods over the usual ones.