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Journal of Global Optimization

, Volume 56, Issue 4, pp 1425–1440 | Cite as

On the determinant and its derivatives of the rank-one corrected generator of a Markov chain on a graph

  • J. A. Filar
  • M. HaythorpeEmail author
  • W. Murray
Article

Abstract

We present an algorithm to find the determinant and its first and second derivatives of a rank-one corrected generator matrix of a doubly stochastic Markov chain. The motivation arises from the fact that the global minimiser of this determinant solves the Hamiltonian cycle problem. It is essential for algorithms that find global minimisers to evaluate both first and second derivatives at every iteration. Potentially the computation of these derivatives could require an overwhelming amount of work since for the Hessian N 2 cofactors are required. We show how the doubly stochastic structure and the properties of the objective may be exploited to calculate all cofactors from a single LU decomposition.

Keywords

Markov chain Generator matrix Derivative Determinant Doubly stochastic LU decomposition Rank-one correction Hamiltonian cycle problem Cofactors 

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References

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Copyright information

© Springer Science+Business Media, LLC. 2012

Authors and Affiliations

  1. 1.Flinders UniversityAdelaideAustralia
  2. 2.Flinders UniversityAdelaideAustralia
  3. 3.Stanford UniversityStanfordUSA

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