MULTISTEP QUASI-NEWTON METHODS FOR OPTIMIZATION

Quasi-Newton methods update, at each iteration, the existing Hessian approximation (or its inverse) by means of data deriving from the step just completed. We show how ''multi-step'' methods (employing, in addition, data from previous iterations) may be constructed by means of interpolating polynomials, leading to a generalization of the ''secant'' (or ''quasi-Newton'') equation. The issue of positive-definiteness in the Hessian approximations is addressed and shown to depend on a generalized version of the condition which is required to hold in the original ''single-step'' methods. The results of extensive numerical experimentation indicate strongly that computational advantages can accrue from such an approach (by comparison with ''single-step'' methods), particularly as the dimension of the problem increases.