In many applications, nodes in a network wish to achieve not only a consensus, but an optimal one. To date, a family of subgradient algorithms have been proposed to solve this problem under general convexity assumptions. This paper shows that, with a few additional mild assumptions, a fundamentally different, non-gradient-based algorithm with appealing features can be constructed. Specifically, we develop Pairwise Equalizing (PE), a gossip-style, distributed asynchronous iterative algorithm for achieving unconstrained, separable, convex consensus optimization over undirected networks with time-varying topologies, where each component function is strictly convex, continuously differentiable, and has a minimizer. We show that PE is easy to implement, bypasses limitations facing the subgradient algorithms, and produces a switched, nonlinear, networked dynamical system that is deterministically and stochastically asymptotically convergent. Moreover, we show that PE admits a common Lyapunov function, reduces to the well-studied Pairwise Averaging and Randomized Gossip Algorithm in a special case, and extends naturally to networks with both time-varying topologies and node memberships.
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