Uncertain parameters of engineering structures can be considered by stochastic distributions (aleatory uncertainty) and intervals or fuzzy numbers (epistemic uncertainty), which leads to a polymorphic uncertain structural response (e.g. probability boxes, fuzzy random variables). For an optimization-based numerical design of structures considering these types of uncertainties, efficient algorithms are required to reduce the computational effort of the nested interval analysis and Monte Carlo simulation (double loop) within the global optimization process. In case of time-consuming finite element models, several simulations with varied parameter combinations can be performed a priori (design of experiment) to train a surrogate model, which reduces the computation time of the global optimization process drastically. However, a global sampling of the training points in the complete range of the design space may lead to an insufficient approximation quality of the objective function close to the optimum. To reduce the approximation error, an adaptive surrogate modeling approach is presented, where additional training data are generated during the optimization process. In all refinement steps, the further search path domain is predicted based on the search path history. New training points are generated to retrain and locally improve the surrogate model. The goals of this approach are to reduce the approximation error close to the optimum and also to reduce the total number of time-consuming finite element simulations. The efficiency of the method is analyzed using different benchmark functions.