ACHIEVEMENT DESCRIPTION A DISTRIBUTED NEWTON METHOD FOR NETWORK UTILITY MAXIMIZATION Ermin Wei, Asu Ozdaglar and Ali Jadbabaie Motivations Summary Distributed Update Matrix splitting Dual update Primal update Increasing interest in distributed optimization and control of ad hoc wireless networks, which are characterized by: Lack of centralized control and access to information Time-varying connectivity Control-optimization algorithms deployed in such networks should be: Distributed relying on local information Robust against changes in the network topology Standard Approach to Distributed Optimization in networks: Use dual decomposition and subgradient (or first-order) methods Yields distributed algorithms for some classes of problems Suffers from slow rate of convergence properties MAIN ACHIEVEMENT: A Newton method that solves network utility maximization problems in a distributed manner Simulations indicate the superiority of the distributed Newton method over dual subgradient methods HOW IT WORKS: Turning inequality constraints into barrier functions Employing matrix splitting techniques on the dual graph to solve the dual Newton step Using a consensus-based local averaging scheme, which requires local information only ASSUMPTIONS AND LIMITATIONS: Routing information and capacity constraints are predetermined Dual and primal steps are computed separately Combine Newton (second order) methods with consensus policies to distribute the computations associated with the dual Newton step IMPACT NEXT-PHASE GOALS ACHIEVEMENT DESCRIPTION STATUS QUO NEW INSIGHTS Most existing distributed optimiza-tion algorithms rely on first order methods • These algorithms are easy to distribute • However, they can be quite slow to converge, limiting their use in rapidly changing dynamic networks Significant improvements with the distributed Newton method compared to subgradient methods Second order methods for distributed network utility maximization Prove convergence and rate of convergence of our methods Understand the impact of network topology on algorithm performance Design algorithms that compute primal and dual steps simultaneously Simulations Simulated over 50 random routing matrices Distributed Newton method is shown to converge much faster than the classic subgradient method Network Topology affects convergence rates of both algorithms similarly (two curves positive correlated) Newton Method NUM Formulation Add slack variables and turn inequalities into logarithm barriers Given an initial primal vector , the iterative update is defined by where is the Newton step update, which solves the following system of equations where is the dual price Conclusion We developed a distributed second order Newton method for network utility maximization problems. Simulations showed significant improvement in convergence speed over standard methods. Future work includes investigation of rate of convergence properties and the effects of network topology on performance.