Convex and Nonconvex Optimization for Control

Abstract

Convex optimization has long been a cornerstone of classical optimal and robust control, enabling strong guarantees and reliable computational tools. At the same time, modern control design increasingly relies on nonconvex optimization, especially direct policy search and learning-based methods, which have shown striking empirical success and are now supported by a growing body of theory.  This talk highlights a unifying perspective that connects these two worlds. We begin by reviewing benign nonconvex landscapes that arise in several benchmark control problems, explaining why simple first-order methods can succeed despite nonconvexity. We then introduce an Extended Convex Lifting (ECL) framework that exposes hidden convexity in classical control from a modern optimization viewpoint. This lifting provides a principled bridge between nonconvex policy optimization and convex reformulations. When an ECL exists, it can (i) transform the original design problem into an equivalent convex program and (ii) certify global optimality for a class of stationary points. We conclude by discussing algorithmic implications and recent scalable methods that make these ideas practical at larger scales. Looking ahead, these connections suggest a promising path to combining the global guarantees of convex optimization with the flexibility of policy optimization for robust and learning-enabled control of modern dynamical systems.