  In many applications, nodes in a network desire 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,
for the scalar case and by assuming a bit more, novel non-gradient-based
algorithms with appealing features can be constructed. Specifically, we develop
Pairwise Equalizing (PE) and Pairwise Bisectioning (PB), two gossip algorithms
that solve unconstrained, separable, convex consensus optimization problems
over undirected networks with time-varying topologies, where each local
function is strictly convex, continuously differentiable, and has a minimizer.
We show that PE and PB are easy to implement, bypass limitations of the
subgradient algorithms, and produce switched, nonlinear, networked dynamical
systems that admit a common Lyapunov function and asymptotically converge.
Moreover, PE generalizes the well-known Pairwise Averaging and Randomized
Gossip Algorithm, while PB relaxes a requirement of PE, allowing nodes to never
share their local functions.
