Distributed Coordination for Separable Convex Optimization with Coupling Constraints

This paper considers a network of agents described by an undirected graph that seek to solve a convex optimization problem with separable objective function and coupling equality and inequality constraints. Both the objective function and the inequality constraints are Lipschitz continuous. We assume that the constraints are compatible with the network topology in the sense that, if the state of an agent is involved in the evaluation of any given local constraint, this agent is able to fully evaluate it with the information provided by its neighbors. Building on the saddle-point dynamics of an augmented Lagrangian function, we develop provably correct distributed continuous-time coordination algorithms that allow each agent to find their component of the optimal solution vector along with the optimal Lagrange multipliers for the equality constraints in which the agent is involved. Our technical approach combines notions and tools from nonsmooth analysis, set-valued and projected dynamical systems, viability theory and convex programming.