The design of the blade-disk interface in a jet engine is of particular importance since the resulting stresses in both disk and blade represent limiting factors for the life of the engine. Hence, it is important to find an optimal design that fulfills lifing requirements and is robust with respect to uncertainties such as manufacturing variabilities. In this paper, the challenges of performing such a robust optimization are analyzed and a solution strategy is proposed. Based on a numerical model of the blade-disk interface, involving a high-dimensional parametrization of the interface geometry, a surrogate-based optimization is carried out. The surrogate model is created from a set of Monte Carlo samples covering the high-dimensional design space. Correlations among the input variables as well as nonlinearities in the numerical model require the use of Gaussian process surrogate models instead of simple polynomial approaches. Based on this, an efficient exploration of the design space is possible, minimizing the relevant stresses of the design. In the next step, the robustness of the optimized model is analyzed via the maximum stresses and the standard deviation as the robust optimum of the design. With this approach, lifing as well as robustness requirements can be incorporated into the design process.