Benign Nonconvex Landscapes in Optimal and Robust Control, Part II: Extended Convex Lifting
<|reference_start|>Benign Nonconvex Landscapes in Optimal and Robust Control, Part II: Extended Convex Lifting: Many optimal and robust control problems are nonconvex and potentially nonsmooth in their policy optimization forms. In Part II of this paper, we introduce a new and unified Extended Convex Lifting (ECL) framework to reveal hidden convexity in classical optimal and robust control problems from a modern optimization perspective. Our ECL offers a bridge between nonconvex policy optimization and convex reformulations, enabling convex analysis for nonconvex problems. Despite non-convexity and non-smoothness, the existence of an ECL not only reveals that minimizing the original function is equivalent to a convex problem but also certifies a class of first-order non-degenerate stationary points to be globally optimal. Therefore, no spurious stationarity exists in the set of non-degenerate policies. This ECL framework can cover many benchmark control problems, including state feedback linear quadratic regulator (LQR), dynamic output feedback linear quadratic Gaussian (LQG) control, and $\mathcal{H}_\infty$ robust control. ECL can also handle a class of distributed control problems when the notion of quadratic invariance (QI) holds. We further show that all static stabilizing policies are non-degenerate for state feedback LQR and $\mathcal{H}_\infty$ control under standard assumptions. We believe that the new ECL framework may be of independent interest for analyzing nonconvex problems beyond control.<|reference_end|>
@article{zheng2024benign,
    title={Benign Nonconvex Landscapes in Optimal and Robust Control, Part II:
  Extended Convex Lifting},
    author={Yang Zheng, Chih-Fan Pai, Yujie Tang},
    journal={arXiv preprint arXiv:2406.04001},
    year={2024},
    archivePrefix={arXiv},
    eprint={2406.04001},
    primaryClass={math.OC cs.SY eess.SY math.DS}
}