In this paper, we show a design method of a feedback control law that achieves output regulation in the presence of actuator saturation. It is well-known that it is impossible to obtain a bounded feedback control law that achieves global output regulation in the case where the plant has unstable poles. For such systems, we show a method of constructing the regulatable region based on the notion of the invariant set theory. Furthermore, we show that we can design a feedback control law that minimizes the LQ type cost function by using the proposed method. The design problem is reduced to a convex optimization problem with linear matrix inequality (LMI) constraints. The effectiveness of the proposed method is shown through a numerical example and an experimental result.