An ADMM algorithm for solving ℓ1 regularized MPC

We present an Alternating Direction Method of Multipliers (ADMM) algorithm for solving optimization problems with an ℓ₁ regularized least-squares cost function subject to recursive equality constraints. The considered optimization problem has applications in control, for example in ℓ₁ regularized MPC. The ADMM algorithm is easy to implement, converges fast to a solution of moderate accuracy, and enables separation of the optimization problem into sub-problems that may be solved in parallel. We show that the most costly step of the proposed ADMM algorithm is equivalent to solving an LQ regulator problem with an extra linear term in the cost function, a problem that can be solved efficiently using a Riccati recursion. We apply the ADMM algorithm to an example of ℓ₁ regularized MPC. The numerical examples confirm fast convergence to sufficient accuracy and a linear complexity in the MPC prediction horizon.