Using function-values in multi-step quasi-Newton methods

In previous work, the authors (1993, 1994) developed the concept of multi-step quasi-Newton methods, based on the use of interpolating polynomials determined by data from the m most recent steps. Different methods for parametrizing these polynomials were studied by the authors (1993), and several methods were shown (empirically) to yield substantial gains over the standard (one-step) BFGS method for unconstrained optimization. In this paper, we will consider the issue of how to incorporate function-value information within the framework of such multi-step methods. This is achieved, in the case of two-step methods, through the use of a carefully chosen rational form to interpolate the three most recent iterates. The results of numerical experiments on the new methods are reported.