  This paper introduces a systematic methodological framework to design and
analyze distributed algorithms for optimization and games over networks.
Starting from a centralized method, we identify an aggregation function
involving all the decision variables (e.g., a global cost gradient or
constraint) and introduce a distributed consensus-oriented scheme to
asymptotically approximate the unavailable information at each agent. Then, we
delineate the proper methodology for intertwining the identified building
blocks, i.e., the optimization-oriented method and the consensus-oriented one.
The key intuition is to interpret the obtained interconnection as a singularly
perturbed system. We rely on this interpretation to provide sufficient
conditions for the building blocks to be successfully connected into a
distributed scheme exhibiting the convergence guarantees of the centralized
algorithm. Finally, we show the potential of our approach by developing a new
distributed scheme for constraint-coupled problems with a linear convergence
rate.
