In this paper a new method for designing optimal ℋ₂/ℋ_∞ dynamic output-feedback controllers is presented. Due to the use of a path-following method, all bilinear matrix inequalities (BMI's), which appear typically in this kind of problems, are converted to linear matrix inequalities (LMI's). The design occurs in two phases. In the first one, a suboptimal, mixed-objective, dynamic output-feedback controller is synthesized. This controller constitutes then the initial step of the computation of an optimal ℋ₂/ℋ_∞ dynamic output-feedback controller, computation which represents the second phase of the design process. The path-following method used in this phase makes it possible to use two different Lyapunov matrices in the two objective constraints, thus reducing the conservatism inherent to other approaches for such a design.