Gradient-free distributed resource allocation via simultaneous perturbation

This note considers a class of decentralized convex optimization problems subject to constraints. We propose a discrete-time algorithm with constant step-size that exploits the simultaneous perturbation method to obtain information of the cost function. Under some technical conditions, we prove practical convergence in probability of the algorithm to a ball that contains the optimizer and which has a step-dependent size. The novelty of our approach is that the agents do not require a closed form expression of the cost function, nor global knowledge of total resources in the network or any specific procedure for algorithm initialization. Our proof methods employ nonsmooth Lyapunov theory, convex analysis, and stochastic difference inclusions. We illustrate the applicability of the algorithm in an electricity market scenario.