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Response surface methods based on kriging and radial basis function (RBF) interpolation have been successfully applied to solve expensive, i.e. com-putationally costly, global black-box nonconvex optimization problems. We describe extensions of these methods to handle linear, nonlinear and integer constraints. In particular standard RBF and new adaptive RBF (ARBF) algorithms are discussed. Test results are presented on standard test problems, both nonconvex problems with linear and nonlinear constraints, and mixed-integer nonlinear problems. Solvers in the TOMLAB Optimization Environment (http://tomopt.com/tomlab/) are compared; the three deterministic derivative-free solvers rbfSolve, ARBFMIP and EGO with three derivative-based mixed-integer nonlinear solvers, OQNLP, MINLPBB and MISQP as well as GENO implementing a stochastic genetic algorithm. Assuming that the objective function is costly to evaluate the performance of the ARBF algorithm proves to be superior.

The optimization environment TOMLAB, http://tomopt.com, has seen a tremendous growth during the last years. Most state-of-the-art optimization software has been hooked up, e.g. KNITRO, SNOPT and CONOPT for large-scale nonlinear programming, and CPLEX and Xpress-MP for large-scale mixed-integer programming. Unique tools for global black-box mixed-integer nonconvex problems have been developed. Originally developed for MATLAB, now TOMLAB is available for LabView as TOMVIEW and .NET as TOMNET. TOMLAB is interfaced with the modelling language AMPL and the DIFFPACK package for advanced PDE solutions. This talk gives an overview over the latest developments.

A mixed-integer constrained extension of the radial basis function (RBF) interpolation algorithm for computationally costly global non-convex optimization is presented. Implementation in TOM-LAB (http://tomlab.biz) solver rbfSolve is discussed. The algorithm relies on mixed-integer nonlinear (MINLP) sub solvers in TOMLAB, e.g. OQNLP, MINLPBB or the constrained DIRECT solvers (glcDirect or glcSolve). Depending on the initial experimental design, the basic RBF algorithm sometimes fails and make no progress. A new method how to detect when there is a problem is presented. We discuss the causes and present a new faster and more robust Adaptive RBF (ARBF) algorithm. Test results for unconstrained problems are discussed.