A continuous-time decentralized optimization scheme with positivity constraints

In this paper we introduce a continuous-time version of a recently proposed decentralized multi-agent optimization scheme. In this scheme, a number of networked agents cooperate in locating the optimum of the sum of their individual objective functions. Each agent has access only to its own objective function and its neighbors´ estimates of the collective optimum. Under mild assumptions, we derive explicit expressions for a lower bound on the algorithm´s convergence rate and an upper bound on the agents´ ultimate estimation error, in terms of relevant problem parameters. We build on the analytic techniques we previously introduced, in which we treat the evolution of the mean and deviation of agents´ estimates as two coupled dynamic subsystems, and provide a Lyapunov argument for the practical asymptotic stability of their interconnection. More generally, this approach turns out to be useful in deriving sharper convergence results under weaker assumptions in the continuous-time case, as well as in providing an elegant way to account for the effects of positive projections that might need to be employed by each agent in some applications. Finally, we propose an application of this scheme to the design of fully decentralized dual resource allocation algorithms.