The proximal augmented Lagrangian method for distributed and embedded nonsmooth composite optimization

在现代技术中，结合了传感、计算和通信设备的大型动力系统网络无处不在。网络系统的主要挑战之一是开发快速且可扩展的方法来进行分析和设计。这样的系统涉及组件的大规模互连，具有快速演变的结构和通信/处理能力的限制，并且需要实时的分布式控制动作。这些要求使得依赖集中信息处理的控制策略变得不可行，并引发了新的最优控制问题。在这些方法中，用典型的非光滑正则化来增强标准性能度量，以促进最优控制器中所需的结构特征(例如，低通信要求)。拟议工作的更广泛影响范围从改善电网的性能和可靠性到系统设计艾滋病毒治疗的联合药物疗法。该提案的教育部分侧重于开发新的非线性和分布式系统课程。该中心将开发新的入门课程，旨在吸引来自不同工程学系的高年级、本科生和一年级研究生。这些课程将强调非线性和网络系统的分析和设计的实际应用、物理解释、结构特征和共同主题。其智力优势在于分布式嵌入式非光滑组合优化理论和技术的发展。结构化的最优控制和逆问题，特别是当试图实时识别和控制快速发展的系统的动态表示时，通常会导致由光滑项和非光滑正则化项之和组成的泛函的优化。PI的最新研究将被用来开发有效和可靠地解决这些问题的理论基础和方法。这一建议的基石是近端增广拉格朗日，这是一个由原始变量和对偶变量组成的连续可微函数，它使非光滑组合优化的一阶和二阶方法得以发展。PI将利用与非光滑正则化子相关联的邻近算子的结构来开发用于大规模分布式和嵌入式优化的高效算法，并使用控制理论工具来确定其收敛速度。这项拟议的工作将为非光滑复合优化提供新的一阶和二阶原始-对偶算法，在面向控制和物理可行的建模方面取得重大进展，并使大型动力系统网络的实时分布式控制成为可能。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Large networks of dynamical systems that combine sensing, computing, and communication devices are ubiquitous in modern technology. One of the major challenges in networked systems is the development of fast and scalable methods for their analysis and design. Such systems involve large-scale interconnections of components, have rapidly-evolving structure and limitations on communication/processing power, and require real-time distributed control actions. These requirements make control strategies that rely on centralized information processing infeasible and motivate new classes of optimal control problems. In these, standard performance metrics are augmented with typically nonsmooth regularizers to promote desired structural features (e.g., low communication requirements) in the optimal controller. The broader impacts of the proposed work range from improved performance and reliability of power grid to systematic design of combination drug therapies for HIV treatment. The educational part of the proposal focuses on the development of new nonlinear and distributed systems curricula. The PI will develop new introductory courses aimed at attracting students from diverse engineering departments at senior undergraduate and first year graduate levels. The courses will emphasize practical applications, physical interpretations, structural features, and common themes in analysis and design of nonlinear and networked systems. The intellectual merit lies in the development of theory and techniques for distributed and embedded nonsmooth composite optimization. Structured optimal control and inverse problems, that arise especially when trying to identify and control dynamical representations of rapidly evolving systems in real-time, typically lead to optimization of functionals consisting of a sum of a smooth term and a nonsmooth regularizer. The PI's recent research will be leveraged to develop theoretical foundation and methods for solving these problems efficiently and reliably. The cornerstone of this proposal is the proximal augmented Lagrangian, a continuously differentiable function of primal and dual variables that enables the development of variety of first and second order methods for nonsmooth composite optimization. The PI will utilize structure of proximal operators associated with nonsmooth regularizers to develop efficient algorithms for large-scale distributed and embedded optimization and employ control-theoretic tools to establish their convergence rates. The proposed effort will furnish new classes of first and second order primal-dual algorithms for nonsmooth composite optimization, lead to significant advances in control-oriented and physically-viable modeling, and enable real-time distributed control of large-scale networks of dynamical 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.