Let the least value of a given function F(x), x in R^n, be required. Many iterative algorithms for this calculation employ quadratic polynomial approximations to F, and, if derivatives are not available, each approximation may be defined by (n + 1)(n + 2)/2 interpolation conditions. Another approach, taken from gradient methods, is to employ fewer conditions, and to take up the freedom in a new quadratic approximation by minimizing a measure of the change from the previous approximation. A technique of this kind will be presented and discussed. It provides some excellent numerical results when the number of interpolation equations is only 2n + 1.