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

• Published in: arXiv.org
• Main Authors: Simpson-Porco, John W, Bala Kameshwar Poolla, Monshizadeh, Nima, Dorfler, Florian
• Format: Paper Journal Article
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 07.11.2018
• Subjects: Algorithms; Augmentation; Computational geometry; Convexity; Design optimization; Economic models; Feedback control; Mathematics - Optimization and Control; Regularization; Resource allocation; Saddle points; Transient performance
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.1803.02182

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.