Benign Nonconvex Landscapes in Optimal and Robust Control, Part II: Extended Convex Lifting

TL;DR

This paper introduces the Extended Convex Lifting framework that reveals hidden convexity in classical control problems, enabling convex analysis and guaranteeing global optimality of certain policies despite nonconvexity.

Contribution

The paper presents a unified ECL framework that transforms nonconvex control problems into convex ones, certifies global optimality of non-degenerate stationary points, and applies to various benchmark control problems.

Findings

ECL reveals hidden convexity in control problems.

Certifies global optimality of non-degenerate stationary points.

Applicable to LQR, LQG, and $\\mathcal{H}_\infty$ control problems.

Abstract

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.