Distributed min-max optimization in networks

We consider a setup where we are given a network of agents with their local objective functions which are coupled through a common decision variable. We provide a distributed stochastic gradient algorithm for the agents to compute an optimal decision variable that minimizes the worst case loss incurred by any agent. We establish almost sure convergence of the agent's estimates to a common optimal point. We demonstrate the use of our algorithm to a problem of min-max fair power allocation in a cellular network.

[1]  Soummya Kar,et al.  Distributed Consensus Algorithms in Sensor Networks With Imperfect Communication: Link Failures and Channel Noise , 2007, IEEE Transactions on Signal Processing.

[2]  Angelia Nedic,et al.  Distributed Stochastic Subgradient Projection Algorithms for Convex Optimization , 2008, J. Optim. Theory Appl..

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

[4]  Michael Athans,et al.  Convergence and asymptotic agreement in distributed decision problems , 1982, 1982 21st IEEE Conference on Decision and Control.

[5]  Srdjan S. Stankovic,et al.  Decentralized Parameter Estimation by Consensus Based Stochastic Approximation , 2007, IEEE Transactions on Automatic Control.

[6]  V. Borkar.  Stochastic Approximation: A Dynamical Systems Viewpoint , 2008 .

[7]  Yuri Ermoliev,et al.  Stochastic quasigradient methods. Numerical techniques for stochastic optimization , 1988 .

[8]  Angelia Nedic,et al.  Distributed Non-Autonomous Power Control through Distributed Convex Optimization , 2009, IEEE INFOCOM 2009.

[9]  D. Donoho.  For most large underdetermined systems of linear equations the minimal 𝓁1‐norm solution is also the sparsest solution , 2006 .

[10]  Mung Chiang,et al.  Power Control in Wireless Cellular Networks , 2008, Found. Trends Netw..

[11]  H. Kushner,et al.  Stochastic Approximation and Recursive Algorithms and Applications , 2003 .

[12]  Slawomir Stanczak,et al.  Unifying view on min-max fairness and utility optimization in cellular networks , 2005, IEEE Wireless Communications and Networking Conference, 2005.

[13]  O. Nelles,et al.  An Introduction to Optimization , 1996, IEEE Antennas and Propagation Magazine.

[14]  Dimitri P. Bertsekas,et al.  Necessary and sufficient conditions for a penalty method to be exact , 1975, Math. Program..