  This paper proposes distributed discrete-time algorithms to cooperatively
solve an additive cost optimization problem in multi-agent networks. The
striking feature lies in the use of only the sign of relative state information
between neighbors, which substantially differentiates our algorithms from
others in the existing literature. We first interpret the proposed algorithms
in terms of the penalty method in optimization theory and then perform
non-asymptotic analysis to study convergence for static network graphs.
Compared with the celebrated distributed subgradient algorithms, which however
use the exact relative state information, the convergence speed is essentially
not affected by the loss of information. We also study how introducing noise
into the relative state information and randomly activated graphs affect the
performance of our algorithms. Finally, we validate the theoretical results on
a class of distributed quantile regression problems.
