Input-Output Performance of Linear-Quadratic Saddle-Point Algorithms with Application to Distributed Resource Allocation Problems

TL;DR

This paper analyzes the input-output performance of saddle-point algorithms for quadratic programs, deriving explicit measures, exploring regularization effects, and proposing distributed methods that match centralized performance in resource allocation tasks.

Contribution

It provides explicit input-output performance metrics for saddle-point algorithms, investigates regularization and augmentation effects, and introduces a distributed dual algorithm with comparable performance.

Findings

Regularization improves transient performance.

Augmentation does not always enhance performance.

Distributed algorithms can match centralized saddle-point methods.

Abstract

Saddle-point or primal-dual methods have recently attracted renewed interest as a systematic technique to design distributed algorithms which solve convex optimization problems. When implemented online for streaming data or as dynamic feedback controllers, these algorithms become subject to disturbances and noise; convergence rates provide incomplete performance information, and quantifying input-output performance becomes more important. We analyze the input-output performance of the continuous-time saddle-point method applied to linearly constrained quadratic programs, providing explicit expressions for the saddle-point H2 norm under a relevant input-output configuration. We then proceed to derive analogous results for regularized and augmented versions of the saddle-point algorithm. We observe some rather peculiar effects -- a modest amount of regularization significantly improves the transient performance, while augmentation does not necessarily offer improvement. We then propose a distributed dual version of the algorithm which overcomes some of the performance limitations imposed by augmentation. Finally, we apply our results to a resource allocation problem to compare the input-output performance of various centralized and distributed saddle-point implementations and show that distributed algorithms may perform as well as their centralized counterparts.