Abstract
We consider a class of weighted gradient methods for distributed resource allocation over a network. Each node of the network is associated with a local variable and a convex cost function; the sum of the variables (resources) across the network is fixed. Starting with a feasible allocation, each node updates its local variable in proportion to the differences between the marginal costs of itself and its neighbors. We focus on how to choose the proportional weights on the edges (scaling factors for the gradient method) to make this distributed algorithm converge and on how to make the convergence as fast as possible.
We give sufficient conditions on the edge weights for the algorithm to converge monotonically to the optimal solution; these conditions have the form of a linear matrix inequality. We give some simple, explicit methods to choose the weights that satisfy these conditions. We derive a guaranteed convergence rate for the algorithm and find the weights that minimize this rate by solving a semidefinite program. Finally, we extend the main results to problems with general equality constraints and problems with block separable objective function.
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Arrow, K. J., and Hurwicz, L., Decentralization and Computation in Resource Allocation, Essays in Economics and Econometrics, Edited by R. W. Pfouts, University of North Carolina Press, Chapel Hill, North Carolina, pp. 34–104, 1960.
Heal, G. M., Planning without Prices, Review of Economic Studies, Vol. 36, pp. 347–362, 1969.
Kurose, J. F., and Simha, R., A Microeconomic Approach to Optimal Resource Allocation in Distributed Computer Systems, IEEE Transactions on Computers, Vol. 38, pp. 705–717, 1989.
Bertsekas, D. P., Nonlinear Programming, 2nd Edition, Athena Scientific, Belmont, Massachusetts, 1999.
Boyd, S., and Vandenberghe, L., Convex Optimization, Cambridge University Press, Cambridge, UK, 2004.
Hurwicz, L., The Design of Mechanisms for Resource Allocation, American Economic Review, Vol. 63, pp. 1–30, 1973.
Ho, Y. C., Servi, L. D., and Suri, R., A Class of Center-Free Resource Allocation Algorithms, Large Scale Systems, Vol. 1, pp. 51–62, 1980.
Servi, L. D., Electrical Networks and Resource Allocation Algorithms, IEEE Transactions on Systems, Man, and Cybernetics, Vol. 10, pp. 841–848, 1980.
Rockafellar, R. T., Network Flows and Monotropic Optimization, John Wiley and Sons, New York, NY, 1984.
Bertsekas, D. P., and Tsttsiklis, J. N., Parallel and Distributed Computation, Prentice-Hall, Englewood, Cliffs, New Jersey, 1989.
Bertsekas, D. P., Network Optimization: Continuous and Discrete Models, Athena Scientific, Belmont, Massachusetts, 1998.
Tsitsiklis, J. N., and Bertsekas, D. P., Distributed Asynchronous Optimal Routing in Data Networks, IEEE Transactions on Automatic Control, Vol. 31, pp. 325–332, 1986.
El Baz, D., Asynchronous Gradient Algorithms for a Class of Convex Separable Network Flow Problems, Computational Optimization and Applications, Vol. 5, pp. 187–205, 1996.
Luo, Z. Q., and Tseng, P., On the Rate of Convergence of a Distributed Asynchronous Routing Algorithm, IEEE Transactions on Automatic Control, Vol. 39, pp. 1123–1129, 1994.
Tsitsiklis, J. N., Bertsekas, D. P., and Athans, M., Distributed Asynchronous Deterministic and Stochastic Gradient Optimization Algorithms, IEEE Transactions on Automatic Control, Vol. 31, pp. 803 812, 1986.
Boyd, S., Diaconis, P., and Xiao, L., Fastest Mixing Markov Chain on a Graph, SIAM Review, Vol. 46, pp. 667–689, 2004.
Xiao, L., and Boyd, S., Fast Linear Iterations for Distributed Averaging, Systems and Control Letters, Vol. 53, pp. 66–78, 2004.
Horn, R. A., and Johnson, C. A., Matrix Analysis, Cambridge University Press, Cambridge, UK, 1985.
Metropolis, N., Rosenbluth, A., Rosenbluth, M., Teller, A., and Teller, E., Equations of State Calculations by Fast Computing Machines, Journal of Chemical Physics, Vol. 21, pp. 1087–1092, 1953.
Diaconis, P., and Saloff-Coste, L., What Do We Know About the Metropolis Algorithms, Journal of Computer and System Sciences, Vol. 57, pp. 20–36, 1998.
Vandenberghe, L., and Boyd, S., Semidefinite Programming, SIAM Review, Vol. 38, pp. 49–95, 1996.
Nesterov, Y., and Nemirovskii, A., Interior-Point Polynomial Algorithms in Convex Programming, SIAM Studies in Applied Mathematics, SIAM, Philadelphia, Pennsylvania, 1994.
Ye, Y., Interior-Point Algorithms: Theory and Analysis, John Wiley and Sons, New York, NY, 1997.
Wolkowicz, H., Saigal, R., and Vandengerghe, L., Editors, Handbook of Semidefinite Programming: Theory, Algorithms, and Applications, Kluwer Academic Publishers, Boston, Massachusetts, 2000.
Merris, R., Laplacian Matrices of Graphs: A Survey, Linear Algebra and Its Applications, Vol. 197, pp. 143–176, 1994.
Godsil, C., and Royle, G., Algebraic Graph Theory, Springer, New York, NY, 2001.
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The authors are grateful to Professor Paul Tseng and the anonymous referee for their valuable comments that helped us to improve the presentation of this paper.
Communicated by P. Tseng
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Xiao, L., Boyd, S. Optimal Scaling of a Gradient Method for Distributed Resource Allocation. J Optim Theory Appl 129, 469–488 (2006). https://doi.org/10.1007/s10957-006-9080-1
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DOI: https://doi.org/10.1007/s10957-006-9080-1