A distributed random projection gradient method for stochastic optimization problems in multi-agent systems

In the context of time-varying directed graphs under uniformly strong connectivity, this paper studies distributed stochastic optimization problems where the local objective functions and inequality constraints are nonlinear, and the local constraint functions are disturbed by uncertain parameters. The goal is to locate a consensus optimal solution to the optimization problem without intervention from a central agent, where each agent obtains the solution through information exchange with neighbouring agents and local computations. In this article, a distributed random projection gradient (DRPG) algorithm is proposed based on the distributed gradient algorithm, random sampling process and Polyak projection operator. Then, this paper theoretically proves that the DRPG algorithm reaches almost sure convergence. Finally, the convergence of the DRPG algorithm is illustrated by a numerical case study.