The design of robust structures requires the consideration of uncertain structural loads and material parameters as well es uncertain design variables. Within the concept of polymorphic uncertainty quantification, stochastic and non-stochastic approaches are fused to consider aleatory and epistemic uncertainties, respectively. In this contribution, it is focused on a combination of random variables and intervals resulting in probability boxes, i.e. imprecise probabilities, to quantify the structural responses. A structural design task can be formulated as an optimization problem, where an objective function has to be minimized by design variables taking constraints into account. Considering polymorphic uncertainties is challenging, because the solution of an optimization problem requires to minimize deterministic objective functions as well as to comply with deterministic constraints. This means that deterministic surrogate measures of uncertain quantities of interest must be defined reflecting the variability and imprecision of structural parameters and design variables. Within robust design optimization approaches, it is focused more on results with low scattering or low imprecision of the objective close to the optima instead of the optimal performance in average. Here, reliability-based design optimization approaches are applied, where statistical measures of the random variables, such as mean values, variances or quantile values, to be optimized or to be constrained are used as deterministic surrogate measures. In a similar way, the interval midpoint, width, and bounds are used as deterministic surrogate measures for intervals. In order to consider not only the robustness but also the general structural performance, an optimization problem with two objectives is obtained, where the compromise solution leads to a pareto front. To solve such optimization problems in structural mechanics, three loops are required, i.e., global optimization, interval analysis, and stochastic analysis of the structural model (e.g. finite element model). The high computational effort is reduced by efficient surrogate models.