Subgradient methods and consensus algorithms for solving convex optimization problems

Abstract

Abstract-In this paper we propose a subgradient method for solving coupled optimization problems in a distributed way given restrictions on the communication topology. The iterative procedure maintains local variables at each node and relies on local subgradient updates in combination with a consensus process. The local subgradient steps are applied simultaneously as opposed to the standard sequential or cyclic procedure. We study convergence properties of the proposed scheme using results from consensus theory and approximate subgradient methods. The framework is illustrated on an optimal distributed finite-time rendezvous problem.

Distributed Asymptotic Minimization of Sequences of Convex Functions by a Broadcast Adaptive Subgradient Method

“…1 Unfortunately, most studies on distributed subgradient methods do not characterize the behavior of the algorithms in such dynamic systems.…”

“…Typically, in these problems centralized approaches are not desirable because of physical limitations (the central agent may not have a direct connection with all other agents) or because of robustness issues (the system may fail if the central agent collapses). Therefore, a great deal of effort has been devoted to the development of non-hierarchical distributed optimization algorithms [1]- [7], [9]- [11]. In particular, here we focus on decentralized subgradient methods where agents can work massively in parallel and exchange information with point-to-multipoint links [1], [5], [6], [9], [11].…”

“…To date, the majority of distributed subgradient methods have focused on static systems where the cost function does not change during the iterations of the algorithm [1], [9], [11]. Formally, once these algorithms start running, agents have to wait until a good estimate of the minimizer of the global function is obtained in every agent.…”

“…Applications that can be posed as such optimization problems include, among others, motion planning in multiagent systems [1], [2], acoustic source localization [3]- [5], and distributed adaptive filtering [6]- [8]. Typically, in these problems centralized approaches are not desirable because of physical limitations (the central agent may not have a direct connection with all other agents) or because of robustness issues (the system may fail if the central agent collapses).…”

“…Therefore, a great deal of effort has been devoted to the development of non-hierarchical distributed optimization algorithms [1]- [7], [9]- [11]. In particular, here we focus on decentralized subgradient methods where agents can work massively in parallel and exchange information with point-to-multipoint links [1], [5], [6], [9], [11]. These approaches often give rise to low-complexity iterative optimization algorithms that are suitable for large-scale systems using a simple communication model among agents.…”