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.

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This paper addresses the problem of solving unconstrained, separable, convex optimization problems over networks and introduces a new approach to the problem: control of distributed convex optimization. We first develop Hopwise Equalizing (HE), a non-gradient-based, distributed asynchronous iterative algorithm that is asymptotically convergent and that is capable of solving the problem. Based on the framework provided by HE, we then develop Controlled Hopwise Equalizing (CHE), showing that a common Lyapunov function, constructed based on the first-order convexity condition, can be used to incorporate the notion of greedy, decentralized, feedback iteration control, whereby individual nodes use potential drops in the value of the Lyapunov function to control, on their own, when to initiate an iteration. Finally, via extensive simulation on wirelessly connected random geometric graphs, we show that CHE is significantly more bandwidth/energy efficient than several existing subgradient algorithms, requiring far less communications to solve a convex optimization problem.

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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 and reduces to the well-studied Pairwise Averaging and Randomized Gossip Algorithm in a special case.

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