Title: Fully Distributed Algorithms for Densely Coupled Optimization Problems in Sparse Optimization and Transportation Applications
Distributed algorithms are gaining increasing attention with broad applications in different areas such as multi-agent network systems, big data, machine learning, and distributed control systems, among others. Most of the distributed optimization algorithms developed assume a separable structure for the underlying optimization problems, and certain coupled optimization problems are often solved via partially distributed schemes. In this thesis, we develop fully distributed algorithms for densely coupled optimization problems in two topics, namely, column partition based sparse optimization problems and transportation applications. Firstly, we develop two-stage, fully distributed algorithms for coupled sparse optimization problems including LASSO, BPDN and their extensions. The proposed algorithms are column partition based and rely on the solution properties, exact regularization, and dual formulation of the problems. The overall convergence of two-stage schemes is shown. Numerical tests demonstrate the effectiveness of the proposed schemes. Secondly, we develop fully distributed algorithms for model predictive control (MPC) based connected and autonomous vehicle (CAV) platooning control under linear and nonlinear vehicle dynamics. In the context of linear vehicle dynamics, the underlying optimization problem of the MPC is a densely coupled, convex quadratically constrained quadratic program (QCQP). A decomposition technique is developed to formulate the densely coupled QCQP as a locally coupled convex optimization problem. We then develop operator splitting method based schemes to solve this problem in a fully distributed manner. Particularly, to meet challenging real-time implementation requirements, a generalized Douglas-Rachford splitting method based distributed algorithm is proposed, along with initial state warm up techniques. Under nonlinear vehicle dynamics, the underlying problem is a densely coupled, nonconvex optimization problem. We develop sequential convex programming based fully distributed optimization algorithms. Control and closed loop stability analysis are carried out for both linear and nonlinear vehicle dynamics. Numerical tests performed for possibly heterogeneous CAV platoons demonstrate the effectiveness of the proposed schemes.