6,200 research outputs found
A Distributed Newton Method for Network Utility Maximization
Most existing work uses dual decomposition and subgradient methods to solve
Network Utility Maximization (NUM) problems in a distributed manner, which
suffer from slow rate of convergence properties. This work develops an
alternative distributed Newton-type fast converging algorithm for solving
network utility maximization problems with self-concordant utility functions.
By using novel matrix splitting techniques, both primal and dual updates for
the Newton step can be computed using iterative schemes in a decentralized
manner with limited information exchange. Similarly, the stepsize can be
obtained via an iterative consensus-based averaging scheme. We show that even
when the Newton direction and the stepsize in our method are computed within
some error (due to finite truncation of the iterative schemes), the resulting
objective function value still converges superlinearly to an explicitly
characterized error neighborhood. Simulation results demonstrate significant
convergence rate improvement of our algorithm relative to the existing
subgradient methods based on dual decomposition.Comment: 27 pages, 4 figures, LIDS report, submitted to CDC 201
Distributed Large Scale Network Utility Maximization
Recent work by Zymnis et al. proposes an efficient primal-dual interior-point
method, using a truncated Newton method, for solving the network utility
maximization (NUM) problem. This method has shown superior performance relative
to the traditional dual-decomposition approach. Other recent work by Bickson et
al. shows how to compute efficiently and distributively the Newton step, which
is the main computational bottleneck of the Newton method, utilizing the
Gaussian belief propagation algorithm.
In the current work, we combine both approaches to create an efficient
distributed algorithm for solving the NUM problem. Unlike the work of Zymnis,
which uses a centralized approach, our new algorithm is easily distributed.
Using an empirical evaluation we show that our new method outperforms previous
approaches, including the truncated Newton method and dual-decomposition
methods. As an additional contribution, this is the first work that evaluates
the performance of the Gaussian belief propagation algorithm vs. the
preconditioned conjugate gradient method, for a large scale problem.Comment: In the International Symposium on Information Theory (ISIT) 200
Distributed Newton-type algorithms for network resource allocation
Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2010.Cataloged from PDF version of thesis.Includes bibliographical references (p. 99-101).Most of today's communication networks are large-scale and comprise of agents with local information and heterogeneous preferences, making centralized control and coordination impractical. This motivated much interest in developing and studying distributed algorithms for network resource allocation problems, such as Internet routing, data collection and processing in sensor networks, and cross-layer communication network design. Existing works on network resource allocation problems rely on using dual decomposition and first-order (gradient or subgradient) methods, which involve simple computations and can be implemented in a distributed manner, yet suffer from slow rate of convergence. Second-order methods are faster, but their direct implementation requires computation intensive matrix inversion operations, which couple information across the network, hence cannot be implemented in a decentralized way. This thesis develops and analyzes Newton-type (second-order) distributed methods for network resource allocation problems. In particular, we focus on two general formulations: Network Utility Maximization (NUM), and network flow cost minimization problems. For NUM problems, we develop a distributed Newton-type fast converging algorithm using the properties of self-concordant utility functions. Our algorithm utilizes novel matrix splitting techniques, which enable both primal and dual Newton steps to be computed using iterative schemes in a decentralized manner with limited information exchange. Moreover, the step-size used in our method can be obtained via an iterative consensus-based averaging scheme. We show that even when the Newton direction and the step-size in our method are computed within some error (due to finite truncation of the iterative schemes), the resulting objective function value still converges superlinearly to an explicitly characterized error neighborhood. Simulation results demonstrate significant convergence rate improvement of our algorithm relative to the existing subgradient methods based on dual decomposition. The second part of the thesis presents a distributed approach based on a Newtontype method for solving network flow cost minimization problems. The key component of our method is to represent the dual Newton direction as the limit of an iterative procedure involving the graph Laplacian, which can be implemented based only on local information. Using standard Lipschitz conditions, we provide analysis for the convergence properties of our algorithm and show that the method converges superlinearly to an explicitly characterized error neighborhood, even when the iterative schemes used for computing the Newton direction and the stepsize are truncated. We also present some simulation results to illustrate the significant performance gains of this method over the subgradient methods currently used.by Ermin Wei.S.M
On Dual Convergence of the Distributed Newton Method for Network Utility Maximization
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.National Science Foundation (U.S.). (Career) (Grant number DMI-0545910)United States. Air Force Office of Scientific Research. Multidisciplinary University Research Initiative (R6756-G2)United States. Office of Naval Research. Multidisciplinary University Research Initiative (Grant N0001408107474)United States. Army Research Office. Multidisciplinary University Research Initiative. ScalableUnited States. Air Force Office of Scientific Research. Complex Networks Progra
A Finite-Time Cutting Plane Algorithm for Distributed Mixed Integer Linear Programming
Many problems of interest for cyber-physical network systems can be
formulated as Mixed Integer Linear Programs in which the constraints are
distributed among the agents. In this paper we propose a distributed algorithm
to solve this class of optimization problems in a peer-to-peer network with no
coordinator and with limited computation and communication capabilities. In the
proposed algorithm, at each communication round, agents solve locally a small
LP, generate suitable cutting planes, namely intersection cuts and cost-based
cuts, and communicate a fixed number of active constraints, i.e., a candidate
optimal basis. We prove that, if the cost is integer, the algorithm converges
to the lexicographically minimal optimal solution in a finite number of
communication rounds. Finally, through numerical computations, we analyze the
algorithm convergence as a function of the network size.Comment: 6 pages, 3 figure
Decomposition by Partial Linearization: Parallel Optimization of Multi-Agent Systems
We propose a novel decomposition framework for the distributed optimization
of general nonconvex sum-utility functions arising naturally in the system
design of wireless multiuser interfering systems. Our main contributions are:
i) the development of the first class of (inexact) Jacobi best-response
algorithms with provable convergence, where all the users simultaneously and
iteratively solve a suitably convexified version of the original sum-utility
optimization problem; ii) the derivation of a general dynamic pricing mechanism
that provides a unified view of existing pricing schemes that are based,
instead, on heuristics; and iii) a framework that can be easily particularized
to well-known applications, giving rise to very efficient practical (Jacobi or
Gauss-Seidel) algorithms that outperform existing adhoc methods proposed for
very specific problems. Interestingly, our framework contains as special cases
well-known gradient algorithms for nonconvex sum-utility problems, and many
blockcoordinate descent schemes for convex functions.Comment: submitted to IEEE Transactions on Signal Processin
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