In nonlinear regression models, standard optimality functions for experimental designs depend on the unknown parameters. An appealing concept for choosing a design is the minimax criterion. By restricting the set of regarded designs in a suitable way, the minimax problem becomes numerically tractable in principle; nevertheless, it is still a two-level problem requiring nested global optimizations. We present a new algorithm which combines successive grid approximations for the parameter space with a technique for reducing the size of the search domain of the outer minimization problem drastically already at the first approximation steps. The method works efficiently for the construction of symmetrical balanced designs in a widely used logistic dose-response model. The solution proposed by the algorithm can be evaluated by a sensitivity analysis.