A Jacobi decomposition algorithm for distributed convex optimization in Distributed Model Predictive Control

In this paper we introduce an iterative distributed Jacobi algorithm for solving convex optimization problems, which is motivated by distributed model predictive control (MPC) for linear time-invariant systems. Starting from a given feasible initial guess, the algorithm iteratively improves the value of the cost function with guaranteed feasible solutions at every iteration step, and is thus suitable for MPC applications in which hard constraints are important. The proposed iterative approach involves solving local optimization problems consisting of only few subsystems, depending on the flexible choice of decomposition and the sparsity structure of the couplings. This makes our approach more applicable to situations where the number of subsystems is large, the coupling is sparse, and local communication is available. We also provide a method for checking a posteriori centralized optimality of the converging solution, using comparison between Lagrange multipliers of the local problems. Furthermore, a theoretical result on convergence to optimality for a particular distributed setting is also provided.