Distributed Convex Optimization with State-Dependent (Social) Interactions over Random Networks

TL;DR

This paper introduces a novel distributed convex optimization method for multi-agent systems with random, state-dependent communication networks, ensuring convergence under broad conditions including asynchronous protocols and periodic graphs.

Contribution

It is the first to analyze both random arbitrary networks and state-dependent interactions, providing a convergent algorithm for a generalized optimization problem over quasi-nonexpansive operators.

Findings

Algorithm converges almost surely and in mean square.

Works with periodic and irreducible weighted graphs.

Applicable to asynchronous and state-dependent network scenarios.

Abstract

This paper aims at distributed multi-agent convex optimization where the communications network among the agents are presented by a random sequence of possibly state-dependent weighted graphs. This is the first work to consider both random arbitrary communication networks and state-dependent interactions among agents. The state-dependent weighted random operator of the graph is shown to be quasi-nonexpansive; this property neglects a priori distribution assumption of random communication topologies to be imposed on the operator. Therefore, it contains more general class of random networks with or without asynchronous protocols. A more general mathematical optimization problem than that addressed in the literature is presented, namely minimization of a convex function over the fixed-value point set of a quasi-nonexpansive random operator. A discrete-time algorithm is provided that is able to converge both almost surely and in mean square to the global solution of the optimization problem. Hence, as a special case, it reduces to a totally asynchronous algorithm for the distributed optimization problem. The algorithm is able to converge even if the weighted matrix of the graph is periodic and irreducible under synchronous protocol. Finally, a case study on a network of robots in an automated warehouse is given where there is distribution dependency among random communication graphs.