  We introduce the Projected Push-Pull algorithm that enables multiple agents
to solve a distributed constrained optimization problem with private cost
functions and global constraints, in a collaborative manner. Our algorithm
employs projected gradient descent to deal with constraints and a lazy update
rule to control the trade-off between the consensus and optimization steps in
the protocol. We prove that our algorithm achieves geometric convergence over
time-varying directed graphs while ensuring that the decision variable always
stays within the constraint set. We derive explicit bounds for step sizes that
guarantee geometric convergence based on the strong-convexity and smoothness of
cost functions, and graph properties. Moreover, we provide additional
theoretical results on the usefulness of lazy updates, revealing the challenges
in the analysis of any gradient tracking method that uses projection operators
in a distributed constrained optimization setting. We validate our theoretical
results with numerical studies over different graph types, showing that our
algorithm achieves geometric convergence empirically.
