A dual Newton strategy for scenario decomposition in robust multistage MPC

Abstract

Summary This paper considers the solution of tree‐structured quadratic programs as they may arise in multistage model predictive control. In this context, sampling the uncertainty on prescribed decision points gives rise to different scenarios that are linked to each other via the so‐called nonanticipativity constraints. Previous work suggests to dualize these constraints and apply Newton's method on the dual problem to achieve a parallelizable scheme. However, it has been observed that the globalization strategy in such an approach can be expensive. To alleviate this problem, we propose to dualize both the nonanticipativity constraints and the dynamics to obtain a computationally cheap globalization. The dual Newton system is then reformulated into small highly structured linear systems that can be solved in parallel to a large extent. The algorithm is complemented by an open‐source software implementation that targets embedded optimal control applications.

A dual Newton strategy for tree‐sparse quadratic programs and its implementation in the open‐source software treeQP

“…The control inputs are forces acting on the first n u masses, with n u < n m . This example is similar to the one used by Kouzoupis et al, with the difference that n x and n u can vary to obtain dynamics of different dimensions.…”

“…In this section, we present in detail a dual decomposition method to solve the tree‐sparse QP in (3). The algorithm (further referred to as Algorithm DN3) is a generalization of the work of Frasch et al (Algorithm DN0) to tree topologies and an alternative approach to the works of Leidereiter et al (Algorithm DN1) and Kouzoupis et al (Algorithm DN2), which also address a robust problem formulation (see Section 4 for details).…”

“…The dual variables are updated in the following fashion: with Δ λ a solution to the (possibly regularized) Newton system and W : = ∇ 2 d ( λ ) + F . The step size t ∈ (0,1] is chosen to enforce convergence, as in the work of Kouzoupis et al, whereas the regularization matrix F ≺0 (typically diagonal to preserve the sparsity structure) ensures that W is strictly negative definite whenever the dual Hessian is rank deficient. Such cases occur when the active local constraints together with the dualized constraints form linear dependencies .…”

“…In this section, we compare the proposed dual Newton scheme (Algorithm DN3) against its predecessor, Algorithm DN2, and against the interior point method of Algorithm IP1 (a recent extension of Algorithm IP0 to tree‐structured problems). All three algorithms are open‐source.…”

“…For a comparison of the proposed scheme against existing approaches, we perform a series of simulations and plot the resulting performance profiles similarly to the numerical experiments in the work of Kouzoupis et al The y‐axis in a performance profile represents the cumulative distribution function of the performance ratio r p , s . The performance ratio is defined as with t p , s denoting the time required for solver s to solve problem p and the set of solvers.…”