Multiple update multi-step methods for unconstrained optimization

The quasi-Newton multi-step methods were developed in [2] and have revealed substantial numerical improvements over the standard single step Secant-based BFGS. Such methods use a variant of the Secant equation that the updated Hessian (or its inverse) satisfies at each iteration. In this paper, we explore algorithms whose updated Hessians satisfy multiple relations of the Secant-type in order that the numerical potentials of such techniques be investigated. We employ a rational model in developing the new methods. The model hosts a free parameter which is exploited in enforcing symmetry on the multi-updated matrix. Our results are encouraging, and the improvements incurred supercede those obtained from other existing methods at minimal extra storage and computational overhead.