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In many applications, nodes in a network desire not only a consensus, but anoptimal one. To date, a family of subgradient algorithms have been proposed tosolve this problem under general convexity assumptions. This paper shows that,for the scalar case and by assuming a bit more, novel non-gradient-basedalgorithms with appealing features can be constructed. Specifically, we developPairwise Equalizing (PE) and Pairwise Bisectioning (PB), two gossip algorithmsthat solve unconstrained, separable, convex consensus optimization problemsover undirected networks with time-varying topologies, where each localfunction is strictly convex, continuously differentiable, and has a minimizer.We show that PE and PB are easy to implement, bypass limitations of thesubgradient algorithms, and produce switched, nonlinear, networked dynamicalsystems that admit a common Lyapunov function and asymptotically converge.Moreover, PE generalizes the well-known Pairwise Averaging and RandomizedGossip Algorithm, while PB relaxes a requirement of PE, allowing nodes to nevershare their local functions.

Gossip Algorithms for Convex Consensus Optimization over Networks