Network Utility Maximization Revisited: Three Issues and Their   Resolution
  Distributed and iterative network utility maximization algorithms, such as the primal-dual algorithms or the network-user decomposition algorithms, often involve trajectories where the iterates may be infeasible, convergence to the optimal points of relaxed problems different from the original, or convergence to local maxima. In this paper, we highlight the three issues with iterative algorithms. We then propose a distributed and iterative algorithm that does not suffer from the three issues. In particular, we assert the feasibility of the algorithm's iterates at all times, convergence to the global maximum of the given problem (rather than to global maximum of a relaxed problem), and avoidance of any associated spurious rest points of the dynamics. A benchmark algorithm due to Kelly, Maulloo and Tan (1998) [Rate control for communication networks: shadow prices, proportional fairness and stability, Journal of the Operational Research Society, 49(3), 237-252] involves fast user updates coupled with slow network updates in the form of additive-increase multiplicative-decrease of suggested user flows. The proposed algorithm may be viewed as one with fast user updates and fast network updates that keeps the iterates feasible at all times. Simulations suggest that the convergence rate of the ordinary differential equation (ODE) tracked by our proposed algorithm's iterates is comparable to that of the ODE for the aforementioned benchmark algorithm. 
Akhil P T and Rajesh Sundaresan
math.OC