SIGEST

Abstract

The SIGEST paper in this issue, “Global Convergence of Radial Basis Function Trust-Region Algorithms for Derivative-Free Optimization,” by Stefan M. Wild and Christine Shoemaker, is an expanded version of “Global Convergence of Radial Basis Function Trust Region Derivative-Free Algorithms,” which appeared in the SIAM Journal on Optimization in 2011. The theoretical contribution of this paper is a convergence proof for a derivative-free optimization method. There are many classes of derivative-free methods. The methods of interest in this paper are for smooth problems where derivatives of the objective function (the function to be minimized) are not available. Such problems are common, arising, for example, from applications where source code is not available. These methods approximate the functions with interpolation, and the challenge for a particular method is using as few interpolatory nodes as possible while simultaneously producing an interpolant which is a useful surrogate for the objective function. The algorithm in this paper uses radial basis functions (RBFs) for the interpolation and manages the convergence of the nonlinear iteration with a trust-region approach. The authors do a fine job of explaining the technical aspects of the algorithm, the nodes for the interpolation, and the convergence analysis. The paper concludes with a compelling application to a subsurface contamination problem. The site is the Blaine Naval Ammunition Depot in Nebraska, and the authors collaborated with a large interdisciplinary team to design a remediation strategy. Three dimensional hydrology codes are used to evaluate the objective function. Because of the internal simulations, the objective function for this problem has noise which arises from internal iterations in the simulators and truncation errors in the discretizations. The RBF trust-region algorithm solved this large and complex design problem very nicely.