부산대학교 도서관

We study a deterministic primal-dual subgradient method for distributed optimization of a separable objective function with global inequality constraints. To control the norm of dual variables, we augment the Lagrangian function with a regularizer on dual variables. Specifically, we show that under a certain restriction on the step size of the underlying algorithm, the norm of dual variables is inversely proportional to the regularizer's curvature. We leverage this result to bound the consensus terms and subsequently establish the convergence rate of the distributed primal-dual algorithm. We also establish an asymptotic decay rate on the norm of the constraint violation. We exhibit a tension between the convergence rate of the underlying algorithm and the decay rate associated with the constraint violation. Specifically, we show that when one of the inequality constraints is binding at optimal solutions, improving the convergence rate in the objective value deteriorates the decay rate of the constraint violation bound and vice versa. However, in the case that one optimal solution satisfies the inequality constraints strictly, the tension vanishes.