In the presence of uncertainty, the optimum obtained based on a nominal identified model can neither provide any performance guarantee nor ensure that critical constraints are satisfied, which is crucial for e.g., bioprocess applications characterized by a high degree of complexity combined with costly experiments. Hence, uncertainty should be considered in the optimization and, furthermore, experiments designed to reduce the uncertainty most important for optimization. Herein, we propose a general framework that combines model-based robust optimization with optimal experiment design. The proposed framework can take advantage of prior knowledge in the form of a mechanistic model structure, and the importance of this is demonstrated by comparing it to more standard black-box models typically employed in learning. Through optimal experiment design, we repeatedly reduce the uncertainty most relevant for optimization so as to maximize the potential for improving the worst-case performance by balancing between exploration and exploitation. This makes the proposed method an efficient model-based robust optimization framework, especially in cases with limited experimental resources. The main part of the paper focuses on the case with modeling uncertainty that can be reduced with the availability of more experimental data. Towards the end of the paper, we consider extending the method to also include inherent uncertainty, such as input uncertainty and unmeasured disturbances. The effectiveness of the method is illustrated through a realistic simulation case study of medium optimization of Chinese hamster ovary cell cultivation in continuous monoclonal antibody production, where the metabolic network consists of 23 extracellular metabolites and 126 reactions.