Improved Strategies for Radial basis Function Methods for Global Optimization

Abstract

We propose some strategies that can be shown to improve the performance of the radial basis function (RBF) method by Gutmann [J. Global optim. 19(3), 201---227 (2001a)] (Gutmann-RBF) and the RBF method by Regis and Shoemaker [J. Global optim. 31, 153---171 (2005)] (CORS---RBF) on some test problems when they are initialized by symmetric Latin hypercube designs (SLHDs). Both methods are designed for the global optimization of computationally expensive functions with multiple local optima. We demonstrate how the original implementation of Gutmann-RBF can sometimes converge slowly to the global minimum on some test problems because of its failure to do local search. We then propose Controlled Gutmann-RBF (CG-RBF), which is a modification of Gutmann-RBF where the function evaluation point in each iteration is restricted to a subregion of the domain centered around a global minimizer of the current RBF model. By varying the size of this subregion in different iterations, we ensure a better balance between local and global search. Moreover, we propose a complete restart strategy for CG-RBF and CORS-RBF whenever the algorithm fails to make any substantial progress after some threshold number of consecutive iterations. Computational experiments on the seven Dixon and Szego [Towards Global optimization, pp. 1---13. North-Holland, Amsterdam (1978)] test problems and on nine Schoen [J. Global optim. 3, 133---137 (1993)] test problems indicate that the proposed strategies yield significantly better performance on some problems. The results also indicate that, for some fixed setting of the restart parameters, the two modified RBF algorithms, namely CG-RBF-Restart and CORS-RBF-Restart, are comparable on the test problems considered. Finally, we examine the sensitivity of CG-RBF-Restart and CORS-RBF-Restart to the restart parameters.