Information Theory for Mobile Ad-Hoc Networks (ITMANET): The FLoWS Project A Distributed Newton Method for Network Optimization Ali Jadbabaie and Asu Ozdaglar

A Distributed Newton Method for Network Optimization (Jadbabaie, Ozdaglar) Distributed Second Order Methods with Convergence Guarantees MAIN ACHIEVEMENT: We developed a Newton method that solves network optimization problems in a distributed manner. We provide convergence and rate of convergence guarantees for the proposed method. Simulation experiments on a series of randomly generated graphs suggest superiority of the distributed Newton method over dual subgradient methods. HOW IT WORKS: Constrained Newton method Dual Newton step found by solving a discrete Poisson equation involving the graph Laplacian. Using a consensus-based local averaging scheme, this can be done using only local information. ASSUMPTIONS AND LIMITATIONS: Solves minimum cost network flow problems Dual and primal steps computed separately Most existing distributed optimiza- tion methods rely on dual decomposition and subgradient (first order) algorithms These algorithms easy to distribute However, they can be quite slow to converge limiting their use in rapidly changing dynamic wireless networks Second order methods for distributed network optimization Understand the impact of network structure (connectivity and mobility) on algorithm performance Design algorithms that compute primal and dual steps jointly Extend second order methods to network utility maximization IMPACT NEXT-PHASE GOALS ACHIEVEMENT DESCRIPTION STATUS QUO NEW INSIGHTS Combine Newton (second order) methods with consensus policies to distribute the computations associated with the dual Newton step Significant improvements with the distributed Newton method compared to subgradient methods

Motivation 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

This Work We propose a new Newton-type (second-order) method, which is distributed and achieves superlinear convergence rate –Relies on representing the dual Newton direction as the solution of a discrete Poisson equation involving the graph Laplacian –Consensus-type iterative schemes used to compute the Newton direction and the stepsize with some error –We show that the proposed method converges superlinearly to an error neighborhood –Simulation results demonstrate the superior performance of our method compared to subgradient schemes

Minimum Cost Network Optimization Problem Consider a network represented by a directed graph Each edge has a convex cost function as a function of the flow on edge e We denote the demand at node i by b i b1b1 bnbn The minimum cost network optimization problem is given by: where A is the node-edge incidence matrix of the graph.

Newton Method Let Given an initial primal vector x 0, the iterates are generated by where v k is the Newton step given as the solution to the following system of linear equations: The dual Newton step w k is given by where H k is the Hessian matrix. This computation requires global information.

Distributed Computation of the Dual Step The key step in developing a decentralized iterative scheme for the computation of the vector w k is to recognize that the matrix is the weighted Laplacian of the underlying graph Hence the computation of dual Newton step can be written as where The preceding equation can be solved iteratively as: For all (dependence on k suppressed for convenience). This iteration relies only on local information

Performance on randomly generated graphs The runtime of the Newton's method significantly less than the subgradient method

Effects of network connectivity on performance Two network topologies: complete and sparse

Conclusions We presented a distributed Newton-type method for minimum cost network optimization problem We used consensus schemes to compute the dual Newton direction and the stepsize in a distributed manner We showed that even in the presence of errors, the proposed method converges superlinearly to an error neighborhood Future Work: –Understand the impact of network structure (connectivity and mobility) on algorithm performance –Ongoing work extends this idea to Network Utility Maximization Papers: –Jadbabaie and Ozdaglar, “A Distributed Newton Method for Network Optimization,” submitted for publication in CDC 2009.