A stochastic augumented Lagrangian algorithm for smooth convex optimization: application to MPC with quadratic constraints - Singh, Nitesh K.; Lupu, Daniela; Necoara, Ion (Mediterranean Conference on Control & Automation, 2025)
In this paper we propose a model predictive control (MPC) scheme for discrete-time linear invariant systems subject to quadratic state constraints and simple input constraints based on a stochastic augmented Lagrangian optimization algorithm. Due to the difficulty of projecting on the primal feasible set of the MPC-based smooth optimization problem, we use the augmented Lagrangian relaxation to handle the complicated constraints and then, we linearize the objective and one randomly chosen functional constraint within the augmented Lagrangian at the current iterate, leading to a stochastic gradient descent update for the primal variables, followed by a random coordinate gradient ascent step to update the dual variables. We provide convergence rates in both optimality and feasibility criteria for the iterates of our optimization algorithm using basic assumptions on the problem. Preliminary numerical experiments on a MPC problem with many quadratic constraints for a multi-machine power system demonstrate the viability and performance of our method when compared to some existing state-of-the-art optimization methods and software.