An ADMM Algorithm for Solving l_1 Regularized MPC

TL;DR

This paper introduces an ADMM algorithm tailored for l_1 regularized MPC problems, offering a simple, fast-converging, and parallelizable solution that efficiently handles recursive constraints and leverages Riccati recursion for key steps.

Contribution

The paper develops a novel ADMM-based method for l_1 regularized MPC, enabling efficient, parallelizable optimization with a Riccati recursion step for improved computational performance.

Findings

Fast convergence to moderate accuracy

Linear complexity in prediction horizon

Effective separation of sub-problems for parallel solving

Abstract

We present an Alternating Direction Method of Multipliers (ADMM) algorithm for solving optimization problems with an l_1 regularized least-squares cost function subject to recursive equality constraints. The considered optimization problem has applications in control, for example in l_1 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 l_1 regularized MPC. The numerical examples confirm fast convergence to moderate accuracy and a linear complexity in the MPC prediction horizon.