LineWalker: Line Search for Black Box Derivative-Free Optimization and Surrogate Model Construction

This paper describes a simple, but effective sampling method for optimizing and learning a discrete approximation (or surrogate) of a multi-dimensional function along a one-dimensional line segment of interest. The method does not rely on derivative information and the function to be learned can be a computationally-expensive ``black box'' function that must be queried via simulation or other means. It is assumed that the underlying function is noise-free and smooth, although the algorithm can still be effective when the underlying function is piecewise smooth. The method constructs a smooth surrogate on a set of equally-spaced grid points by evaluating the true function at a sparse set of judiciously chosen grid points. At each iteration, the surrogate's non-tabu local minima and maxima are identified as candidates for sampling. Tabu search constructs are also used to promote diversification. If no non-tabu extrema are identified, a simple exploration step is taken by sampling the midpoint of the largest unexplored interval. The algorithm continues until a user-defined function evaluation limit is reached. Numerous examples are shown to illustrate the algorithm's efficacy and superiority relative to state-of-the-art methods, including Bayesian optimization and NOMAD, on primarily nonconvex test functions.