On the construction of minimisation methods of quasi-Newton type

The secant equation, which underlies all standard ‘quasi-Newton’ minimisation methods, arises from the use of a linear function to model the gradient along a chosen direction. The authors present new minimisation algorithms, derived by replacing this linear model with a more general one involving a free parameter, which is determined by using information contained in the current approximate Hessian. The use of such a model can give more flexibility in the criteria to be satisfied during the line-search. The new methods can operate as soon as a reasonable approximation to the Hessian has been accumulated and may, in one sense, be viewed as acceleration techniques for quasi-Newton methods.