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

University Of Southern California, Los Angeles CA

Investigators

Abstract

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.