1. 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.