A primal-dual laplacian gradient flow dynamics for distributed resource allocation problems

Abstract

We employ the proximal augmented Lagrangian method to solve a class of convex resource allocation problems over a connected undirected network of n agents. The agents are coupled by a linear resource equality constraint and their states are confined to a nonnegative orthant. By introducing the indicator function associated with a nonnegative orthant, we bring the problem into a composite form with a nonsmooth objective and linear equality constraints. A primal-dual Laplacian gradient flow dynamics based on the proximal augmented Lagrangian is proposed to solve the problem in a distributed way. These dynamics conserve the sum of the agent states and the corresponding equilibrium points are the KarushKuhn-Tucker points of the original problem. We combine a Lyapunov-based argument with LaSalle's invariance principle to establish global asymptotic stability and use an economic dispatch case study to demonstrate the effectiveness of the proposed algorithm.