Staffer Day Template

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

• 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
Significant improvements with the distributed
Newton method compared to subgradient methods

• 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

Combine Newton (second order) methods with
consensus policies to distribute the computations
associated with the dual Newton step
Distributed Second Order Methods with Convergence
Guarantees
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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

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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

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Minimum Cost Network Optimization Problem
b1

• 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 bi

bn

The minimum cost network optimization problem is
given by
where A is the node-edge incidence matrix of the
graph.
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Newton Method

• Let
• Given an initial primal vector x0, the iterates
  are generated by
• where vk is the Newton step given as the
  solution to the following system of linear
  equations
• The dual Newton step wk is given by
• where Hk is the Hessian matrix.
• This computation requires global information.

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Distributed Computation of the Dual Step

• The key step in developing a decentralized
  iterative scheme for the computation of the
  vector wk 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

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Performance on randomly generated graphs

• The runtime of the Newton's method significantly
  less than the subgradient method

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Effects of network connectivity on
performanceTwo network topologies complete and
sparse
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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.