  We consider a distributed multi-agent optimization problem over a
time-invariant undirected graph, where each agent possesses a local objective
function and all agents collaboratively minimize the average of all objective
functions through local computations and communications among neighbors.
Recently, a class of distributed gradient methods has been proposed that
achieves both exact and geometric convergence when a constant step size is
used. The geometric convergence of these methods is ensured for conservatively
selected step sizes, but how to choose an appropriate step size while running
the algorithms has not been fully addressed. The Barzilai-Borwein (BB) method
is a simple and effective technique for step sizes and requires few storage and
inexpensive computations. It has been widely applied in various areas. In this
paper, we introduce the BB method to distributed optimization. Based on an
adapt-then-combine variation of the dynamic average consensus approach and
using multi-consensus inner loops, we propose a distributed gradient method
with BB step sizes (DGM-BB-C). Our method computes the step size for each agent
automatically which only depends on its local information and is independent of
that for other agents, and the larger step sizes are always permissible. Our
method can seek the exact optimum when the number of consensus steps stays
constant. We prove that DGM-BB-C has geometric convergence to the optimal
solution. Simulation results on a distributed sensing problem show that our
method is superior to some advanced methods in terms of iterations, gradient
evaluations, communications and the related cost framework. These results
validate our theoretical discoveries.
