  In this paper, we consider the decentralized dynamic optimization problem
defined over a multi-agent network. Each agent possesses a time-varying local
objective function, and all agents aim to collaboratively track the drifting
global optimal solution that minimizes the summation of all local objective
functions. The decentralized dynamic optimization problem can be solved by
primal or primal-dual methods, and when the problem degenerates to be static,
it has been proved in literature that primal-dual methods are superior to
primal ones. This motivates us to ask: are primal-dual methods necessarily
better than primal ones in decentralized dynamic optimization?
  To answer this question, we investigate and compare convergence properties of
the primal method, diffusion, and the primal-dual approach, decentralized
gradient tracking (DGT). Theoretical analysis reveals that diffusion can
outperform DGT in certain dynamic settings. We find that DGT and diffusion are
significantly affected by the drifts and the magnitudes of optimal gradients,
respectively. In addition, we show that DGT is more sensitive to the network
topology, and a badly-connected network can greatly deteriorate its convergence
performance. These conclusions provide guidelines on how to choose proper
dynamic algorithms in various application scenarios. Numerical experiments are
constructed to validate the theoretical analysis.
