An optimal first-order primal-dual gap reduction framework for constrained convex optimization

We introduce an analysis framework for constructing optimal first-order primal-dual methods for the prototypical constrained convex optimization template. While this class of methods offers scalability advantages in obtaining numerical solutions, they have the disadvantage of producing sequences that are only approximately feasible to the problem constraints. As a result, it is theoretically challenging to compare the efficiency of different methods. To this end, we rigorously prove in the worst-case that the convergence of primal objective residual in first-order primal-dual algorithms must compete with their constraint feasibility convergence, and mathematically summarize this fundamental trade-off. We then provide a heuristic-free analysis recipe for constructing optimal first-order primal-dual algorithms that can obtain a desirable trade-off between the primal objective residual and feasibility gap and whose iteration convergence rates cannot be improved. Our technique obtains a smoothed estimate of the primal-dual gap and drives the smoothness parameters to zero while simultaneously minimizing the smoothed gap using problem first-order oracles.