Dual decomposition for QPs in scenario tree NMPC

In the field of nonlinear model predictive control (NMPC) under uncertainty we use a robustification approach based on a scenario tree formulation. This approach is known to be less conservative than worst-case approaches. A main challenge of scenario tree NMPC in high-dimensional uncertainty spaces is the exponential growth of the number of scenarios for possible parameter realizations. Hence, the solution of the resulting optimization problem in every NMPC iteration becomes a bottleneck for the computation. We have to solve the problem with fast numerical methods to ensure real-time applicability of scenario tree NMPC. By a multiple shooting discretization we discretize the optimal control problems of an NMPC iteration to obtain a sequence of Nonlinear Programs that we solve by a real-time feasible variant of Sequential Quadratic Programming (SQP). Every single Quadratic Programming problem (QP) exhibits a particular structure originating from the scenario tree. We use a dual decomposition approach on the non-anticipativity constraints from the scenario tree formulation to decouple the large-scale QP into many smaller QPs that each correspond to one scenario in the scenario tree. Within an outer level non-smooth Newton iteration in the dual space of the coupling constraints, the decoupled scenario QPs can be solved in a massively parallel fashion. In the final part of this contribution we present numerical results for NMPC with large scenario trees.