On the conditioning of the Hessian approximation in quasi-Newton methods

AbstractA bound on the possible deterioration in the condition number of the inverse Hessian approximation is derived for updating formulae of BFGS-type. We construct a nonlinear model for the gradient of the objective function along a chosen ray in the variable-space. The model has a “free” parameter which is determined by minimising the condition number bound. Numerical tests indicate that using such a device on the first iteration can lead to an improvement in the performance of several algorithms of quasi-Newton type