In this paper we consider a general problem set-up for a wide class of convex
and robust distributed optimization problems in peer-to-peer networks. In this
set-up convex constraint sets are distributed to the network processors who
have to compute the optimizer of a linear cost function subject to the
constraints. We propose a novel fully distributed algorithm, named
cutting-plane consensus, to solve the problem, based on an outer polyhedral
approximation of the constraint sets. Processors running the algorithm compute
and exchange linear approximations of their locally feasible sets.
Independently of the number of processors in the network, each processor stores
only a small number of linear constraints, making the algorithm scalable to
large networks. The cutting-plane consensus algorithm is presented and analyzed
for the general framework. Specifically, we prove that all processors running
the algorithm agree on an optimizer of the global problem, and that the
algorithm is tolerant to node and link failures as long as network connectivity
is preserved. Then, the cutting plane consensus algorithm is specified to three
different classes of distributed optimization problems, namely (i) inequality
constrained problems, (ii) robust optimization problems, and (iii) almost
separable optimization problems with separable objective functions and coupling
constraints. For each one of these problem classes we solve a concrete problem
that can be expressed in that framework and present computational results. That
is, we show how to solve: position estimation in wireless sensor networks, a
distributed robust linear program and, a distributed microgrid control problem.Comment: submitted to IEEE Transactions on Automatic Contro
