Model-based nonlinear optimal experimental design (OED) has advanced to a relevant systematic tool in model development. However, the high nonlinearity of OED problems and their required high computational effort makes the optimization a challenging task. The objective of this work is to point out a crucial aspect of OED problems, namely, the existence of local minima. For this purpose, general characteristics of OED problems are presented and discussed, using different OED criteria and approaches (sequential/parallel). Furthermore, five different approaches are compared to overcome the local minima problem. Tuning options to improve the computation time of hybrid approaches are derived and discussed. The approaches are compared, based on the results of two case studies. To solve high-dimensional complex OED problems, a proposed hybrid framework is found to be a promising approach.