Input/output analysis of primal-dual gradient algorithms

The primal-dual or saddle-point gradient algorithm has recently attracted interest as a systematic technique for solving distributed optimization problems. To examine the robustness of the algorithm, here we introduce exogenous disturbance inputs and quantify the performance of the algorithm in terms of the induced L
₂
-gain from the disturbance to deviations around the optimizer. For convex problems without inequality constraints, we find that the L
₂
-gain from a disturbance to the deviation of the primal state from the optimizer depends only on how strongly convex the agent objective functions are, and not on the equality constraints or on algorithm time constants. For primal-dual laws derived from an augmented Lagrangian, we show that the L
₂
-gain is a non-increasing function of the augmentation parameters, and therefore that augmentation may be beneficial for improving input/output performance.