Global optimization of costly nonconvex functions using radial basis functions

The paper considers global optimization of costly objective functions, i.e. the problem of finding the global minimum when there are several local minima and each function value takes considerable CPU time to compute. Such problems often arise in industrial and financial applications, where a function value could be a result of a time-consuming computer simulation or optimization. Derivatives are most often hard to obtain, and the algorithms presented make no use of such information. Several algorithms to handle the global optimization problem are described, but the emphasis is on a new method by Gutmann and Powell, a radial basis function method for global optimization. This method is a response surface method, similar to the Efficient Global Optimization method of Jones. Our Matlab implementation of the Radial Basis Function (RBF) method is described in detail and we analyze its efficiency on the standard test problem set of Dixon–Szegö, as well as its applicability on a real life industrial problem from train design optimization. The results show that our implementation of the RBF algorithm is very efficient on the standard test problems compared to other known solvers, but even more interesting, it performs extremely well on the train design optimization problem.