Accelerated dual descent for network optimization

Dual descent methods are commonly used to solve network optimization problems because their implementation can be distributed through the network. However, their convergence rates are typically very slow. This paper introduces a family of dual descent algorithms that use approximate Newton directions to accelerate the convergence rate of conventional dual descent. These approximate directions can be computed using local information exchanges thereby retaining the benefits of distributed implementations. The approximate Newton directions are obtained through matrix splitting techniques and sparse Taylor approximations of the inverse Hessian. We show that, similarly to conventional Newton methods, the proposed algorithm exhibits superlinear convergence within a neighborhood of the optimal value. Numerical analysis corroborates that convergence times are between one to two orders of magnitude faster than existing distributed optimization methods.

[1]  Dimitri P. Bertsekas,et al.  Network optimization : continuous and discrete models , 1998 .

[2]  Naum Zuselevich Shor,et al.  Minimization Methods for Non-Differentiable Functions , 1985, Springer Series in Computational Mathematics.

[3]  Jean Charles Gilbert,et al.  Numerical Optimization: Theoretical and Practical Aspects , 2003 .

[4]  Mark E. J. Newman,et al.  The Structure and Function of Complex Networks , 2003, SIAM Rev..

[5]  Alejandro Ribeiro,et al.  A distributed line search for network optimization , 2012, 2012 American Control Conference (ACC).

[6]  Duncan J. Watts,et al.  Collective dynamics of ‘small-world’ networks , 1998, Nature.

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

[8]  Alejandro Ribeiro,et al.  Network optimization under uncertainty , 2012, 2012 IEEE 51st IEEE Conference on Decision and Control (CDC).

[9]  Derong Liu.  The Mathematics of Internet Congestion Control , 2005, IEEE Transactions on Automatic Control.

[10]  D. Bertsekas,et al.  Distributed asynchronous relaxation methods for convex network flow problems , 1987 .

[11]  Alejandro Ribeiro,et al.  Separation Principles in Wireless Networking , 2010, IEEE Transactions on Information Theory.

[12]  R. Olfati-Saber.  Ultrafast consensus in small-world networks , 2005, Proceedings of the 2005, American Control Conference, 2005..

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

[14]  Asuman E. Ozdaglar,et al.  Distributed Subgradient Methods for Convex Optimization Over Random Networks , 2011, IEEE Transactions on Automatic Control.

[15]  Marc Teboulle,et al.  A Fast Iterative Shrinkage-Thresholding Algorithm for Linear Inverse Problems , 2009, SIAM J. Imaging Sci..

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

[17]  Asuman E. Ozdaglar,et al.  Subgradient methods in network resource allocation: Rate analysis , 2008, 2008 42nd Annual Conference on Information Sciences and Systems.

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

[19]  Rayadurgam Srikant,et al.  The Mathematics of Internet Congestion Control , 2003 .

[20]  Alejandro Ribeiro,et al.  Ergodic Stochastic Optimization Algorithms for Wireless Communication and Networking , 2010, IEEE Transactions on Signal Processing.

[21]  Vladimir Kolmogorov,et al.  New algorithms for the dual of the convex cost network flow problem with application to computer vision , 2007 .

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

[23]  Fan Chung,et al.  Spectral Graph Theory , 1996 .

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

[25]  A. Banerjee.  Convex Analysis and Optimization , 2006 .

[26]  M..  Projected Newton Methods and Optimization of Multicommodity Flows , 2022 .

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

[28]  Asuman E. Ozdaglar,et al.  Approximate Primal Solutions and Rate Analysis for Dual Subgradient Methods , 2008, SIAM J. Optim..

[29]  Alejandro Ribeiro,et al.  Robust Routing in Wireless Multi-Hop Networks , 2007, 2007 41st Annual Conference on Information Sciences and Systems.

[30]  A. Odlyzko,et al.  Bounds for eigenvalues of certain stochastic matrices , 1981 .

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

[32]  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.

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

[34]  J. Sun,et al.  Applying a Newton Method to Strictly Convex Separable Network Quadratic Programs , 1998, SIAM J. Optim..

[35]  James B. Orlin,et al.  Minimum Convex Cost Dynamic Network Flows , 1984, Math. Oper. Res..

[36]  Robert E. Tarjan,et al.  Network Flow Algorithms , 1989 .

[37]  Stephen P. Boyd,et al.  Simultaneous routing and resource allocation via dual decomposition , 2004, IEEE Transactions on Communications.

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

[39]  K. Schittkowski,et al.  NONLINEAR PROGRAMMING , 2022 .

[40]  Asuman E. Ozdaglar,et al.  A fast distributed proximal-gradient method , 2012, 2012 50th Annual Allerton Conference on Communication, Control, and Computing (Allerton).

[41]  C. Kelley.  Solving Nonlinear Equations with Newton's Method , 1987 .

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

[43]  Leandros Tassiulas,et al.  Stability properties of constrained queueing systems and scheduling policies for maximum throughput in multihop radio networks , 1992 .

[44]  Asuman E. Ozdaglar,et al.  A distributed Newton method for Network Utility Maximization , 2010, 49th IEEE Conference on Decision and Control (CDC).