  We study finite-time performance of a recently proposed distributed dual
subgradient (DDSG) method for convex constrained multi-agent optimization
problems. The algorithm enjoys performance guarantees on the last primal
iterate, as opposed to those derived for ergodic means for vanilla DDSG
algorithms. Our work improves the recently published convergence rate of
$\Ocal(\log T/\sqrt{T})$ with decaying step-sizes to $\Ocal(1/\sqrt{T})$ with
constant step-size on a metric that combines suboptimality and constraint
violation. We then numerically evaluate the algorithm on three grid
optimization problems. Namely, these are tie-line scheduling in multi-area
power systems, coordination of distributed energy resources in radial
distribution networks, and joint dispatch of transmission and distribution
assets. The DDSG algorithm applies to each problem with various relaxations and
linearizations of the power flow equations. The numerical experiments
illustrate various properties of the DDSG algorithm--comparison with vanilla
DDSG, impact of the number of agents, and why Nesterov-style acceleration fails
in DDSG settings.
