Penalty and Augmented Lagrangian Methods for Model Predictive Control

Automation will undoubtedly grow to become the cornerstone of various industries, such as manufacturing, automatic transportation, agriculture, and household appliances. Novel challenging control applications continue to emerge, leading to increased interest in advanced control methodologies, such as the optimization based model predictive control (MPC), since they possess inherent capability to include complex objectives and constraints in their problem formulation. The integration of MPC in practice has not been without roadblocks, however. Challenges arise both in modeling complex scenarios and solving the resulting optimization problems in real-time. This thesis uses penalty and augmented Lagrangian approaches, a class of strategies for constrained optimization, to make headway in both of these areas.

In a first part, this thesis considers the application of autonomous navigation in environments with obstacles of general shapes. Previous research in optimization based autonomous navigation is restricted to circular, rectangular, polygonal or obstacles of convex shape. In contrast, we consider obstacles of which the boundaries are defined by smooth functions, allowing for much more generality in the shapes. Although the resulting constraints are nonsmooth, this thesis shows how a quadratic penalty method is theoretically sound and extremely efficient in practice in solving such problems.

In a second part, this thesis constructs a novel quadratic programming (QP) solver, QPALM, based on the proximal augmented Lagrangian method. The inner minimization is tailored to QPs by making use of efficient semismooth Newton directions and optimal step sizes. Research on QP solvers is typically restricted to convex QPs. In contrast, we also consider the possibility of finding stationary points of nonconvex QPs, relying only on minor modifications of the algorithm. QPALM is shown to theoretically exhibit a linear (outer) convergence rate, even for nonconvex QPs, and demonstrated to possess a unique combination of robustness and efficiency when compared to state-of-the-art QP solvers.