A CLASS OF MODIFIED BFGS METHODS WITH FUNCTION VALUE INFORMATION FOR UNCONSTRAINED OPTIMIZATION

Abstract

Based on some new interpolation conditions, a quadratic interpolation model is constructed to approximate the objective function, and then a class of modified BFGS methods with function value information is presented. The new methods satisfy some new weak secant equations and there is a parameter γ in the update formulae which ranges from zero to one. The global and local superlinear convergence properties of the new modified BFGS methods are proved. Numerical results for standard test problems from CUTE are reported, which indicate that all the methods in the proposed class perform well. Ensuring the sufficient positive definiteness of the updating matrices, an adaptive BFGS quasi-Newton method by dynamically choosing the parameter γ is proposed, which may be competitive with other BFGS modifications.