We use the dual decomposition method along with the dual subgradient algorithm to decouple the linear quadratic optimal control problem for a system of single-integrator vehicles. This produces the optimal control law in a localized manner, in the sense that vehicles can iteratively compute their primal and dual variables by only communicating with their immediate neighbors. In particular, we demonstrate that each vehicle only needs to receive the primal variable of the vehicle ahead and the dual variable of the vehicle behind. We then assume a structured feedback gain relationship between the state and actuation signals, and reformulate the optimization problem to find the optimal feedback gains. We develop an algorithm whereby vehicles can compute structured feedback gains in a localized manner. Convergence properties of the latter algorithm are improved by employing a relaxed version of the augmented Lagrangian method, and numerical examples are provided to demonstrate the utility of our results.