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 scalar 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.

[1]  J. Gillis,et al.  Matrix Iterative Analysis , 1961 .

[2]  R. Varga.  Geršgorin And His Circles , 2004 .

[3]  B. Mohar.  Some applications of Laplace eigenvalues of graphs , 1997 .

[4]  Gerald G. Brown,et al.  Design and Implementation of Large-Scale Primal Transshipment Algorithms , 1976 .

[5]  Steven H. Low,et al.  Optimization flow control—I: basic algorithm and convergence , 1999, TNET.

[6]  John N. Tsitsiklis,et al.  Convergence Speed in Distributed Consensus and Averaging , 2009, SIAM J. Control. Optim..

[7]  Jie Lin,et al.  Coordination of groups of mobile autonomous agents using nearest neighbor rules , 2003, IEEE Trans. Autom. Control..

[8]  Yonina C. Eldar,et al.  Convex Optimization in Signal Processing and Communications , 2009 .

[9]  Steven H. Low,et al.  Optimization flow control with Newton‐like algorithm , 2000, Telecommun. Syst..

[10]  Ιωαννησ Τσιτσικλησ,et al.  PROBLEMS IN DECENTRALIZED DECISION MAKING AND COMPUTATION , 1984 .

[11]  Frank Kelly,et al.  Charging and rate control for elastic traffic , 1997, Eur. Trans. Telecommun..

[12]  Rayadurgam Srikant,et al.  The Mathematics of Internet Congestion Control (Systems and Control: Foundations and Applications) , 2004 .

[13]  Anthony V. Fiacco,et al.  Nonlinear programming;: Sequential unconstrained minimization techniques , 1968 .

[14]  Stephen P. Boyd,et al.  Distributed large scale network utility maximization , 2009, 2009 IEEE International Symposium on Information Theory.

[15]  Asuman E. Ozdaglar,et al.  Distributed Subgradient Methods for Multi-Agent Optimization , 2009, IEEE Transactions on Automatic Control.

[16]  Roland W. Freund,et al.  A QMR-based interior-point algorithm for solving linear programs , 1997, Math. Program..

[17]  Asuman E. Ozdaglar,et al.  Convergence rate for consensus with delays , 2010, J. Glob. Optim..

[18]  U. Feige,et al.  Spectral Graph Theory , 2015 .

[19]  Stephen P. Boyd,et al.  Convex Optimization , 2004, Algorithms and Theory of Computation Handbook.

[20]  J.N. Tsitsiklis,et al.  Convergence in Multiagent Coordination, Consensus, and Flocking , 2005, Proceedings of the 44th IEEE Conference on Decision and Control.

[21]  C. Kelley.  Iterative Methods for Linear and Nonlinear Equations , 1987 .

[22]  Frank Kelly,et al.  Rate control for communication networks: shadow prices, proportional fairness and stability , 1998, J. Oper. Res. Soc..

[23]  A. Robert Calderbank,et al.  Layering as Optimization Decomposition: A Mathematical Theory of Network Architectures , 2007, Proceedings of the IEEE.

[24]  Asuman Ozdaglar,et al.  Cooperative distributed multi-agent optimization , 2010, Convex Optimization in Signal Processing and Communications.

[25]  Asuman E. Ozdaglar,et al.  A distributed newton method for network optimization , 2009, Proceedings of the 48h IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese Control Conference.

[26]  Richard W. Cottle,et al.  Linear Complementarity Problem. , 1992 .

[27]  Audra E. Kosh,et al.  Linear Algebra and its Applications , 1992 .

[28]  Richard M. Murray,et al.  Consensus problems in networks of agents with switching topology and time-delays , 2004, IEEE Transactions on Automatic Control.

[29]  D. Bertsekas,et al.  Projected Newton methods and optimization of multicommodity flows , 1982, 1982 21st IEEE Conference on Decision and Control.

[30]  Dimitri P. Bertsekas,et al.  Convex Analysis and Optimization , 2003 .

[31]  John G. Klincewicz,et al.  A Newton method for convex separable network flow problems , 1983, Networks.

[32]  M. Saunders,et al.  Solution of Sparse Indefinite Systems of Linear Equations , 1975 .

[33]  Yousef Saad,et al.  Iterative methods for sparse linear systems , 2003 .

[34]  Dimitri P. Bertsekas,et al.  Nonlinear Programming , 1997 .

[35]  Yurii Nesterov,et al.  Interior-point polynomial algorithms in convex programming , 1994, Siam studies in applied mathematics.

[36]  M. J. D. Powell,et al.  Nonlinear Programming—Sequential Unconstrained Minimization Techniques , 1969 .

[37]  R. Dembo,et al.  INEXACT NEWTON METHODS , 1982 .

[38]  N. Biggs.  Algebraic Graph Theory , 1974 .

[39]  John N. Tsitsiklis,et al.  Distributed Asynchronous Deterministic and Stochastic Gradient Optimization Algorithms , 1984, 1984 American Control Conference.

[40]  F. Jarre.  Interior-point methods for convex programming , 1992 .

[41]  Asuman E. Ozdaglar,et al.  On dual convergence of the distributed Newton method for Network Utility Maximization , 2011, IEEE Conference on Decision and Control and European Control Conference.

[42]  Convex Optimization in Signal Processing and Communications , 2010 .

[43]  Robert J. Plemmons,et al.  Nonnegative Matrices in the Mathematical Sciences , 1979, Classics in Applied Mathematics.

[44]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[45]  Dimitri P. Bertsekas,et al.  Convex Optimization Theory , 2009 .