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On Dual Convergence of the Distributed Newton Method for Network Utility Maximization

Author(s)
Wei, Ermin; Zargham, Michael; Ozdaglar, Asuman E.; Jadbabaie, Ali
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Abstract
The existing distributed algorithms for Network Utility Maximization (NUM) problems mostly rely on dual decomposition and first-order (gradient or subgradient) methods, which suffer from slow rate of convergence. Recent works [17] and [18] proposed an alternative distributed Newton-type second-order algorithm for solving NUM problems with self-concordant utility functions. This algorithm is implemented in the primal space and involves for each primal iteration computing the dual variables using a finitely terminated iterative scheme obtained through novel matrix splitting techniques. These works presented a convergence rate analysis for the primal iterations and showed that if the error level in the Newton direction (resulting from finite termination of dual iterations) is below a certain threshold, then the algorithm achieves local quadratic convergence rate to an error neighborhood of the optimal solution. This paper builds on these works and presents a convergence rate analysis for the dual iterations that enables us to explicitly compute at each primal iteration the number of dual steps that can satisfy the error level. This yields for the first time a fully distributed second order method for NUM problems with local quadratic convergence guarantee. Simulation results demonstrate significant convergence rate improvement of our algorithm, even when only one dual update is implemented per primal iteration, relative to the existing first-order methods based on dual decomposition.
Date issued
2011-12
URI
http://hdl.handle.net/1721.1/72677
Department
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
Journal
Proceedings on the 50th IEEE Conference on Decision and Control and European Control Conference (CDC-ECC), 2011
Publisher
Institute of Electrical and Electronics Engineers (IEEE)
Citation
Wei, Ermin et al. “On Dual Convergence of the Distributed Newton Method for Network Utility Maximization.” 50th IEEE Conference on Decision and Control and European Control Conference (CDC-ECC), 2011. 6612–6617.
Version: Author's final manuscript
ISBN
978-1-61284-799-3
978-1-61284-800-6
ISSN
0743-1546

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