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Distributed Optimization Algorithms on Structurally Balanced Signed Networks

In this paper, we consider the distributed optimization problem under structurally balanced signed graph. First, we convert the original distributed optimization problem into a conditional minimum problem under the condition that the graph is structurally balanced. Our goal is to find the saddle points of augmented Lagrange function. Inspired by the Lagrange multiplier method, we present our algorithms for both undirected graph and digraph, and show that our algorithms asymptotically converge to the global minimizer. Particularly, our algorithms for digraph can not only handle the weight balanced case but the weight unbalanced case. We show that the unsigned graph is a special case of our signed graph cases. Finally, theoretical results are illustrated by numerical simulations.