Improved Convergence Rates for Distributed Resource Allocation

In this paper, we develop a class of decentralized algorithms for solving a convex resource allocation problem in a network of $n$ agents, where the agent objectives are decoupled while the resource constraints are coupled. The agents communicate over a connected undirected graph, and they want to collaboratively determine a solution to the overall network problem, while each agent only communicates with its neighbors. We first study the connection between the decentralized resource allocation problem and the decentralized consensus optimization problem. Then, using a class of algorithms for solving consensus optimization problems, we propose a novel class of decentralized schemes for solving resource allocation problems in a distributed manner. Specifically, we first propose an algorithm for solving the resource allocation problem with an $o(1/k)$ convergence rate guarantee when the agents' objective functions are generally convex (could be nondifferentiable) and per agent local convex constraints are allowed; We then propose a gradient-based algorithm for solving the resource allocation problem when per agent local constraints are absent and show that such scheme can achieve geometric rate when the objective functions are strongly convex and have Lipschitz continuous gradients. We have also provided scalability/network dependency analysis. Based on these two algorithms, we have further proposed a gradient projection-based algorithm which can handle smooth objective and simple constraints more efficiently. Numerical experiments demonstrates the viability and performance of all the proposed algorithms

Fixed-time Distributed Optimization under Time-Varying Communication Topology

This paper presents a method to solve distributed optimization problem within a fixed time over a time-varying communication topology. Each agent in the network can access its private objective function, while exchange of local information is permitted between the neighbors. This study investigates first nonlinear protocol for achieving distributed optimization for time-varying communication topology within a fixed time independent of the initial conditions. For the case when the global objective function is strictly convex, a second-order Hessian based approach is developed for achieving fixed-time convergence. In the special case of strongly convex global objective function, it is shown that the requirement to transmit Hessians can be relaxed and an equivalent first-order method is developed for achieving fixed-time convergence to global optimum. Results are further extended to the case where the underlying team objective function, possibly non-convex, satisfies only the Polyak-\L ojasiewicz (PL) inequality, which is a relaxation of strong convexity.Comment: 25 page

Persistence based analysis of consensus protocols for dynamic graph networks

This article deals with the consensus problem involving agents with time-varying singularities in the dynamics or communication in undirected graph networks. Existing results provide control laws which guarantee asymptotic consensus. These results are based on the analysis of a system switching between piecewise constant and time-invariant dynamics. This work introduces a new analysis technique relying upon classical notions of persistence of excitation to study the convergence properties of the time-varying multi-agent dynamics. Since the individual edge weights pass through singularities and vary with time, the closed-loop dynamics consists of a non-autonomous linear system. Instead of simplifying to a piecewise continuous switched system as in literature, smooth variations in edge weights are allowed, albeit assuming an underlying persistence condition which characterizes sufficient inter-agent communication to reach consensus. The consensus task is converted to edge-agreement in order to study a stabilization problem to which classical persistence based results apply. The new technique allows precise computation of the rate of convergence to the consensus value.Comment: This article contains 7 pages and includes 4 figures. it is accepted in 13th European Control Conferenc