CAREER: Interplay between Convex and Nonconvex Optimization for Control

University Of California-San Diego, La Jolla CA

Investigators

Abstract

Feedback is a fundamental principle underlying many self-regulating natural and technological systems. Control as the principled use of feedback loops and algorithms has been deeply embedded in many engineering systems, including aerospace, energy, transportation, healthcare, and robotic systems. This CAREER project will tackle fundamental challenges related to the interplay between convex and nonconvex optimization for modern control systems. On the one hand, convex reformulations or relaxations have gained popularity in control, thanks to advances in interior-point algorithms. While these methods often provide rigorous stability and safety certificates, their applicability tends to be limited to individual dynamic systems or centralized settings. On the other hand, the old topic of policy search for directly optimiz to empirical successes of reinforcement learning. This class of methods is conceptually simpler, computationally more flexible, but leads to nonconvex optimization, making it harder to derive theoretical guarantees. This project will establish theoretical and algorithmic foundations for bridging convex and nonconvex optimization for a broader class of modern control systems. The outcomes will significantly broaden the optimization and control problems in societal engineering systems, including transportation, power grids, and smart buildings, facilitating efficient and reliable solutions. The project tightly integrates comprehensive educational and outreach activities. A suite of curriculum materials for control education will be developed, lowering the barrier to understanding fundamental feedback principles. The project team will lead activities in summer training camps and collaborate with well-established programs at UCSD, contributing to knowledge dissemination to the general public and K-12 students. This project consists of three synergistic thrusts, fully investigating the interplay between convex and nonconvex optimization for modern control. First, we will develop an innovative framework to reveal hidden convexity in nonconvex static and dynamic distributed control of networked systems. Our framework will advance closed-loop convexity, sparsity invariance, and efficient formulations of linear matrix inequalities. Second, we will establish theoretical guarantees and algorithmic foundations for nonconvex policy search. Specifically, we will develop convex lifting analysis to certify global optimality in smooth nonconvex optimal control and establish algorithmic foundations for nonsmooth and nonconvex robust control with robustness and safety requirements. Lastly, we will develop scalable convex and nonconvex optimization algorithms for large-scale systems by leveraging nonsmooth eigenvalue optimization, decomposition, and acceleration schemes. Collectively, these advances will result in new foundational theoretical frameworks and practical scalable algorithms to achieve reliable and efficient modern control of societal engineering systems. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.