CAREER: Interplay between Convex and Nonconvex Optimization for Control

反馈是许多自我调节的自然和技术系统的基本原则。控制作为反馈回路和算法的原则使用，已经深深嵌入到许多工程系统中，包括航空航天、能源、交通、医疗保健和机器人系统。这个CAREER项目将解决与现代控制系统的凸和非凸优化之间相互作用相关的基本挑战。一方面，由于内点算法的进步，凸重构或松弛在控制中得到了普及。虽然这些方法通常提供严格的稳定性和安全性证书，但它们的适用性往往限于单个动态系统或集中设置。另一方面，旧的策略搜索主题直接优化到强化学习的经验成功。这类方法在概念上更简单，在计算上更灵活，但会导致非凸优化，从而更难获得理论保证。该项目将为更广泛的现代控制系统建立桥接凸和非凸优化的理论和算法基础。其成果将显著拓宽社会工程系统（包括交通、电网和智能建筑）的优化和控制问题，促进高效可靠的解决方案。该项目紧密结合了全面的教育和推广活动。将开发一套控制教育的课程材料，降低理解基本反馈原则的障碍。项目团队将领导夏令营活动，并与加州大学圣地亚哥分校的成熟项目合作，为公众和K-12学生的知识传播做出贡献。该项目由三个协同推力组成，充分研究现代控制中凸和非凸优化之间的相互作用。首先，我们将开发一个创新的框架来揭示网络系统的非凸静态和动态分布式控制中的隐藏凸性。我们的框架将推进闭环凸性、稀疏不变性和线性矩阵不等式的有效公式。第二，建立非凸策略搜索的理论保证和算法基础。具体来说，我们将发展凸提升分析来证明光滑非凸最优控制的全局最优性，并为具有鲁棒性和安全性要求的非光滑和非凸鲁棒控制建立算法基础。最后，我们将通过利用非光滑特征值优化、分解和加速方案，为大规模系统开发可扩展的凸和非凸优化算法。总的来说，这些进步将导致新的基础理论框架和实用的可扩展算法，以实现可靠和有效的社会工程系统的现代控制。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.