Control approaches to distributed optimization and network problems

Abstract

Solving a large-scale optimization problem in a distributed manner has been drawing research attention in recent years. This thesis focuses on designing continuous-time dynamical algorithms to solve distributed optimization problems and addressing some challenging problems arising from large-scale distributed networks with control methodologies. Specifically, this thesis addresses commonly encountered problems in large-scale networks, including coupled inequality constraints, unknown and heterogeneous communication delays, directed and jointly strongly connected communication, fully distributed design, and discrete-time communication.

This thesis is mainly divided into two parts. The first part is on the inequality constraints in distributed optimization. To remove the barrier of non-smooth analysis for the discontinuous dynamics caused by inequalities, a generalized Lagrange multiplier method (GLMM) is introduced, which is also capable of solving inequality constraints that are coupled among local agents. The dynamics derived from the GLMM is without projection operators and hence smooth, whose convergence can be rigorously proved by Lyapunov stability theory. Moreover, it is shown that the algorithmic dynamic satisfies passivity properties. Based on the passivity properties, the distributed constrained optimization problem with general convex objective functions and unknown heterogeneous communication delays is addressed by incorporating the control techniques of phase lead compensator and scattering transformation into the algorithmic dynamics.

The second part is on the directed and jointly connected communication topologies. To this end, output consensus by output feedback interconnections among a group of agents that are characterized by passivity indices is addressed first. Then, a distributed algorithm, whose error system with respect to the optimal point can be decomposed into a group of individual input feedforward passive (IFP) systems that interact with each other using diffusive couplings, is proposed. Based on this output feedback interconnection model of IFP agents, this thesis studies the convergence of the algorithm over uniformly jointly strongly connected (UJSC) digraphs, addresses periodic and event-triggered communication mechanisms, discretization of continuous-time algorithms, and proposes novel fully distributed algorithms through passivity techniques and Lyapunov stability theory.