Abstract: We investigate the optimal leader-follower consensus problem in heterogeneous multi-agent systems, which consist of one static leader, first-order followers, and second-order followers. Under conditions in which the position digraph contains a spanning tree and at least one second-order follower takes the static leader as a neighbor, we present the necessary and sufficient condition for this system to achieve optimal consensus by employing the linear-quadratic regulator (LQR) theory and the inverse optimization method. In this sense, the proposed control protocols for each follower are optimized with respect to the obtained linear-quadratic-performance index function. The simulation results confirm the validity of the proposed method.