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.