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

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 $\mathcal {H}_2$ 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

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

ISSN:0018-9286ISSN:1558-252

Control of Average and Deviation in Large-Scale Linear Networks

International audienceThis paper deals with the problem of controlling the average state of a large-scale linear network to a constant reference value. We design an output-feedback controller such that no information about state vector or system matrices is needed. For this controller to have arbitrary positive gains, it is sufficient that only a sign condition on system matrices should be satisfied. To assure that the states of the network are close to the average state, the problem of deviation minimization is solved in addition, using a novel extremum seeking algorithm