  System Level Synthesis (SLS) enables improved robust MPC formulations by
allowing for joint optimization of the nominal trajectory and controller. This
paper introduces a tailored algorithm for solving the corresponding disturbance
feedback optimization problem. The proposed algorithm builds on a recently
proposed joint optimization scheme and iterates between optimizing the
controller and the nominal trajectory while converging q-linearly to an optimal
solution. We show that the controller optimization can be solved through
Riccati recursions leading to a horizon-length, state, and input scalability of
$\mathcal{O}(N^2 ( n_x^3 + n_u ^3 ) )$ for each iterate. On a numerical
example, the proposed algorithm exhibits computational speedups of order $10$
to $10^3$ compared to general-purpose commercial solvers.
