Stochastic subgradient methods with approximate Lagrange multipliers

We study the use of approximate Lagrange multipliers in the stochastic subgradient method for the dual problem in constrained convex optimisation. The use of approximate Lagrange multipliers in the optimisation (instead of the true multipliers) is motivated by the fact that it is possible to accurately approximate some non-convex control problems as convex optimisations. For example, it is possible to solve certain stochastic discrete decision problems by solving a sequence of convex optimisations. We show how the analysis can be used in networking problems with queues, and present a wireless example that has constraints on how control actions can be selected which illustrates the power of the approach.

[1]  R. Srikant,et al.  Fair resource allocation in wireless networks using queue-length-based scheduling and congestion control , 2007, TNET.

[2]  Eytan Modiano,et al.  Dynamic Server Allocation Over Time-Varying Channels With Switchover Delay , 2012, IEEE Transactions on Information Theory.

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

[4]  Douglas J. Leith,et al.  On the relationship between queues and multipliers , 2014, 2014 52nd Annual Allerton Conference on Communication, Control, and Computing (Allerton).

[5]  Longbo Huang,et al.  Delay reduction via Lagrange multipliers in stochastic network optimization , 2011, IEEE Trans. Autom. Control..

[6]  Sean P. Meyn.  Control Techniques for Complex Networks: Workload , 2007 .

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

[8]  D. Leith,et al.  Descent With Approximate Multipliers is Enough: Generalising Max-Weight , 2015, 1511.02517.

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

[10]  Michael J. Neely,et al.  Optimal Energy and Delay Tradeoffs for Multiuser Wireless Downlinks , 2007, IEEE Transactions on Information Theory.

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

[12]  Douglas J. Leith,et al.  Max-Weight Revisited: Sequences of Nonconvex Optimizations Solving Convex Optimizations , 2016, IEEE/ACM Transactions on Networking.

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

[14]  Alexander L. Stolyar,et al.  Maximizing Queueing Network Utility Subject to Stability: Greedy Primal-Dual Algorithm , 2005, Queueing Syst. Theory Appl..

[15]  Leandros Tassiulas,et al.  Stability properties of constrained queueing systems and scheduling policies for maximum throughput in multihop radio networks , 1990, 29th IEEE Conference on Decision and Control.