Distributed Convex Optimization with State-Dependent Interactions over Random Networks

In this paper, an unconstrained collaborative optimization of a sum of convex functions is considered where agents make decisions using local information from their neighbors. The communication between nodes are described by a random sequence of possibly state-dependent weighted networks. It is shown that the state-dependent weighted random operator of the graph has quasi-nonexpansivity property, and therefore the operator does not need the distribution of random communication topologies. Hence, it includes random networks with/without asynchronous protocols. As an extension of the problem, a more general mathematical optimization problem than that of the literature is defined, namely minimization of a convex function over the fixed-value point set of a quasi-nonexpansive random operator. A discrete-time algorithm using diminishing step size is given which can converge almost surely to the global solution of the optimization problem under suitable assumptions. Consequently, as a special case, the algorithm reduces to a totally asynchronous algorithm without requiring distribution dependency or B-connectivity assumption for the distributed optimization problem. The algorithm still works in the case where weighted matrix of the graph is periodic and irreducible in a synchronous protocol.