An algorithm for efficient optimization directed by noisy data

At the heart of several efficient nonlinear-optimization algorithms is a parabolic fit in which the function (or a one-dimensional "slice" of its hypersurface) is represented by three evaluations in the neighborhood of a local extremum. (The extremum of the parabola through those three points is assumed to approximate the function's true extremum.) The three-point solution fails (optimization converges slowly or not at all) when function evaluations contain enough noise, granularity, etc. The algorithm presented here solves analytically for the best-fit parabola and line(s) to four or more data points, calculates the uncertainties of fit parameters, compares chi-square values of the fits, and uses other heuristic information to direct the search for an optimal value.