In this paper, the distributed optimal formation control problem of heterogeneous Euler–Lagrange multi-agent systems with generic formation constraints and inequality constraints is investigated. Based on the primal–dual dynamics and the adaptive control technique, a distributed optimal formation controller consists of a velocity reference signal generator and a velocity tracking controller is proposed. By using the optimality condition, the relationship between the equilibrium point of the closed-loop system and the optimal solution of the optimization problem is established. Then, by utilizing Lyapunov stability analysis, it is rigorously proved that the optimal formation is reached with the proposed controller. Lastly, simulation examples are provided to substantiate the theoretical results.