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Volumn 46, Issue 4, 2004, Pages 667-689

Fastest mixing Markov chain on a graph

Author keywords

Fast mixing; Markov chains; Second largest eigenvalue modulus; Semidefinite programming; Subgradient method

Indexed keywords

ALGORITHMS; EIGENVALUES AND EIGENFUNCTIONS; GRADIENT METHODS; GRAPH THEORY; HEURISTIC METHODS; LAGRANGE MULTIPLIERS; MATHEMATICAL PROGRAMMING; MATRIX ALGEBRA; NUMERICAL METHODS; OPTIMIZATION; POLYNOMIALS; PROBABILITY DISTRIBUTIONS;

EID: 11344275321     PISSN: 00361445     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0036144503423264     Document Type: Article
Times cited : (655)

References (57)
  • 1
    • 0000569649 scopus 로고
    • Random walk on finite groups and rapidly mixing Markov chains
    • Lecture Notes in Math. 986, Springer-Verlag, New York
    • D. Aldous, Random walk on finite groups and rapidly mixing Markov chains, in Séminaire de Probabilités XVII, Lecture Notes in Math. 986, Springer-Verlag, New York, 1983, pp. 243-297.
    • (1983) Séminaire de Probabilités XVII , pp. 243-297
    • Aldous, D.1
  • 2
    • 0003169116 scopus 로고
    • Interior point methods in semidefinite programming with applications to combinatorial optimization
    • F. Alizadeh, Interior point methods in semidefinite programming with applications to combinatorial optimization, Siam J. Optim., 5 (1995), pp. 13-51.
    • (1995) Siam J. Optim. , vol.5 , pp. 13-51
    • Alizadeh, F.1
  • 4
    • 0242710969 scopus 로고    scopus 로고
    • Introduction to Markov Chains, with Special Emphasis on Rapid Mixing
    • Vieweg, Weisbaden, Germany
    • E. Behrends, Introduction to Markov Chains, with Special Emphasis on Rapid Mixing, Adv. Lectures Math., Vieweg, Weisbaden, Germany, 2000.
    • (2000) Adv. Lectures Math.
    • Behrends, E.1
  • 6
    • 0039772219 scopus 로고    scopus 로고
    • Solving large-scale sparse semidefinite programs for combinatorial optimization
    • S. J. Benson, Y. Ye, and X. Zhang, Solving large-scale sparse semidefinite programs for combinatorial optimization, Siam J. Optim., 10 (2000), pp. 443-461.
    • (2000) Siam J. Optim. , vol.10 , pp. 443-461
    • Benson, S.J.1    Ye, Y.2    Zhang, X.3
  • 8
    • 0041386088 scopus 로고    scopus 로고
    • A geometric interpretation of the Metropolis-Hastings algorithm
    • L. Billera and P. Diaconis, A geometric interpretation of the Metropolis-Hastings algorithm, Statist. Sci., 16 (2001), pp. 335-339.
    • (2001) Statist. Sci. , vol.16 , pp. 335-339
    • Billera, L.1    Diaconis, P.2
  • 10
    • 29844451677 scopus 로고    scopus 로고
    • Symmetry analysis of reversible Markov chains
    • to appear; available online from
    • S. Boyd, P. Diaconis, P. A. Parrilo, and L. Xiao, Symmetry analysis of reversible Markov chains, Internet Math., to appear; available online from http://www.stanford.edu/~boyd/symmetry.html.
    • Internet Math.
    • Boyd, S.1    Diaconis, P.2    Parrilo, P.A.3    Xiao, L.4
  • 11
    • 11344264478 scopus 로고    scopus 로고
    • Fastest mixing Markov chain on a path
    • to appear; available online from
    • S. Boyd, P. Diaconis, J. Sun, and L. Xiao, Fastest mixing Markov chain on a path, Amer. Math. Monthly, to appear; available online from http://www.stanford.edu/~boyd/fmmc_path.html.
    • Amer. Math. Monthly
    • Boyd, S.1    Diaconis, P.2    Sun, J.3    Xiao, L.4
  • 12
    • 0004055894 scopus 로고    scopus 로고
    • Cambridge University Press, Cambridge, UK, available online from
    • S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press, Cambridge, UK, 2004; available online from http:// www.stanford.edu/~boyd/cvxbook.html.
    • (2004) Convex Optimization
    • Boyd, S.1    Vandenberghe, L.2
  • 13
    • 0003618624 scopus 로고    scopus 로고
    • Markov Chains, Gibbs Fields, Monte Carlo Simulation and Queues
    • Springer-Verlag, Berlin, Heidelberg
    • P. Brémaud, Markov Chains, Gibbs Fields, Monte Carlo Simulation and Queues, Texts Appl. Math., Springer-Verlag, Berlin, Heidelberg, 1999.
    • (1999) Texts Appl. Math.
    • Brémaud, P.1
  • 15
    • 0016624432 scopus 로고
    • The minimization of certain nondifferentiable sums of eigenvalues of symmetric matrices
    • J. Cullum, W. E. Donath, and P. Wolfe, The minimization of certain nondifferentiable sums of eigenvalues of symmetric matrices, Math. Programming Stud., 3 (1975), pp. 35-55.
    • (1975) Math. Programming Stud. , vol.3 , pp. 35-55
    • Cullum, J.1    Donath, W.E.2    Wolfe, P.3
  • 17
    • 0001874271 scopus 로고
    • Rectangular arrays with fixed margins
    • D. Aldous et al., eds., Springer-Verlag, New York
    • P. Diaconis and A. Gangolli, Rectangular arrays with fixed margins in Discrete Probability and Algorithms, D. Aldous et al., eds., Springer-Verlag, New York, 1995, pp. 15-42.
    • (1995) Discrete Probability and Algorithms , pp. 15-42
    • Diaconis, P.1    Gangolli, A.2
  • 18
    • 0030487036 scopus 로고    scopus 로고
    • Logarithmic Sobolev inequalities for finite Markov chains
    • P. Diaconis and L. Saloff-Coste, Logarithmic Sobolev inequalities for finite Markov chains, Ann. Appl. Probab., 6 (1996), pp. 695-750.
    • (1996) Ann. Appl. Probab. , vol.6 , pp. 695-750
    • Diaconis, P.1    Saloff-Coste, L.2
  • 19
    • 0032136909 scopus 로고    scopus 로고
    • What do we know about the Metropolis algorithms?
    • P. Diaconis and L. Saloff-Coste, What do we know about the Metropolis algorithms?, J. Comput. System Sci., 57 (1998), pp. 20-36.
    • (1998) J. Comput. System Sci. , vol.57 , pp. 20-36
    • Diaconis, P.1    Saloff-Coste, L.2
  • 20
    • 0002740591 scopus 로고
    • Geometric bounds for eigenvalues of Markov chains
    • P. Diaconis and D. Stroock, Geometric bounds for eigenvalues of Markov chains, Ann. Appl. Probab., 1 (1991), pp. 36-61.
    • (1991) Ann. Appl. Probab. , vol.1 , pp. 36-61
    • Diaconis, P.1    Stroock, D.2
  • 21
    • 0032387555 scopus 로고    scopus 로고
    • Algebraic algorithms for sampling from conditional distributions
    • P. Diaconis and B. Sturmfels, Algebraic algorithms for sampling from conditional distributions, Ann. Statist., 26 (1998), pp. 363-397.
    • (1998) Ann. Statist. , vol.26 , pp. 363-397
    • Diaconis, P.1    Sturmfels, B.2
  • 22
    • 84947796534 scopus 로고    scopus 로고
    • Engineering diffusive load balancing algorithms using experiments
    • Lecture Notes in Comput. Sci. 1253, Springer-Verlag, Berlin
    • R. Diekmann, S. Muthukrishnan, and M. V. Nayakkankuppam, Engineering diffusive load balancing algorithms using experiments, in Solving Irregularly Structured Problems in Parallel, Lecture Notes in Comput. Sci. 1253, Springer-Verlag, Berlin, 1997, pp. 111-122.
    • (1997) Solving Irregularly Structured Problems in Parallel , pp. 111-122
    • Diekmann, R.1    Muthukrishnan, S.2    Nayakkankuppam, M.V.3
  • 23
    • 0002236459 scopus 로고
    • Logarithmic Sobolev inequalities and contractivity properties of semigroups
    • Lecture Notes in Math. 1563, Springer-Verlag, Berlin
    • L. Gross, Logarithmic Sobolev inequalities and contractivity properties of semigroups, in Dirichlet Forms (Varenna, 1992), Lecture Notes in Math. 1563, Springer-Verlag, Berlin, 1993, pp. 54-88.
    • (1993) Dirichlet Forms (Varenna, 1992) , pp. 54-88
    • Gross, L.1
  • 24
    • 77956890234 scopus 로고
    • Monte Carlo sampling methods using Markov chains and their applications
    • W. Hastings, Monte Carlo sampling methods using Markov chains and their applications, Biometrika, 57 (1970), pp. 97-109.
    • (1970) Biometrika , vol.57 , pp. 97-109
    • Hastings, W.1
  • 25
    • 0034415006 scopus 로고    scopus 로고
    • A spectral bundle method for semidefinite programming
    • C. Helmberg and F. Rendl, A spectral bundle method for semidefinite programming, Siam J. Optim., 10 (2000), pp. 673-696.
    • (2000) Siam J. Optim. , vol.10 , pp. 673-696
    • Helmberg, C.1    Rendl, F.2
  • 27
    • 0004151494 scopus 로고
    • Cambridge University Press, Cambridge, UK
    • R. A. Horn and C. A. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, UK, 1985.
    • (1985) Matrix Analysis
    • Horn, R.A.1    Johnson, C.A.2
  • 29
    • 0024904712 scopus 로고
    • Approximating the permanent
    • M. Jerrum and A. Sinclair, Approximating the permanent, Siam J. Comput., 18 (1989), pp. 1149-1178.
    • (1989) Siam J. Comput. , vol.18 , pp. 1149-1178
    • Jerrum, M.1    Sinclair, A.2
  • 30
    • 84947909878 scopus 로고    scopus 로고
    • A semidefinite bound for mixing tares of Markov chains
    • Lecture Notes in Comput. Sci. 1084, Springer-Verlag, Berlin
    • N. Kahale, A semidefinite bound for mixing tares of Markov chains, in Proceedings of the 5th Integer Programming and Combinatorial Optimization Conference, Lecture Notes in Comput. Sci. 1084, Springer-Verlag, Berlin, 1996, pp. 190-203.
    • (1996) Proceedings of the 5th Integer Programming and Combinatorial Optimization Conference , pp. 190-203
    • Kahale, N.1
  • 32
    • 0030304775 scopus 로고    scopus 로고
    • Convex analysis on the Hermitian matrices
    • A. S. Lewis, Convex analysis on the Hermitian matrices, Siam J. Optim., 6 (1996), pp. 164-177.
    • (1996) Siam J. Optim , vol.6 , pp. 164-177
    • Lewis, A.S.1
  • 33
    • 0003111058 scopus 로고    scopus 로고
    • Nonsmooth analysis of eigenvalues
    • A. S. Lewis, Nonsmooth analysis of eigenvalues, Math. Programming, 84 (1999), pp. 1-24.
    • (1999) Math. Programming , vol.84 , pp. 1-24
    • Lewis, A.S.1
  • 34
    • 0042602157 scopus 로고    scopus 로고
    • Eigenvalue optimization
    • A. S. Lewis and M. L. Overton, Eigenvalue optimization, Acta Numer., 5 (1996), pp. 149-190.
    • (1996) Acta Numer. , vol.5 , pp. 149-190
    • Lewis, A.S.1    Overton, M.L.2
  • 35
    • 0004182828 scopus 로고    scopus 로고
    • Monte Carlo Strategies in Scientific Computing
    • Springer Ser. Statist., Springer-Verlag, New York
    • J. Liu, Monte Carlo Strategies in Scientific Computing, Springer Ser. Statist., Springer-Verlag, New York, 2001.
    • (2001)
    • Liu, J.1
  • 37
    • 0034547467 scopus 로고    scopus 로고
    • A bundle method for efficiently solving large structured linear matrix inequalities
    • Chicago, IL
    • S. A. Miller and R. S. Smith, A bundle method for efficiently solving large structured linear matrix inequalities, in Proceedings of the American Control Conference, Chicago, IL, 2000, pp. 1405-1409.
    • (2000) Proceedings of the American Control Conference , pp. 1405-1409
    • Miller, S.A.1    Smith, R.S.2
  • 38
    • 0003254248 scopus 로고
    • Interior-Point Polynomial Algorithms in Convex Programming
    • Siam, Philadelphia
    • Y. Nesterov and A. Nemirovskii, Interior-Point Polynomial Algorithms in Convex Programming, Siam Stud. Appl. Math., Siam, Philadelphia, 1994.
    • (1994) Siam Stud. Appl. Math.
    • Nesterov, Y.1    Nemirovskii, A.2
  • 39
    • 0003181133 scopus 로고
    • Large-scale optimization of eigenvalues
    • M. L. Overton, Large-scale optimization of eigenvalues, Siam J. Optim., 2 (1992), pp. 88-120.
    • (1992) Siam J. Optim. , vol.2 , pp. 88-120
    • Overton, M.L.1
  • 40
    • 34250080273 scopus 로고
    • Optimality conditions and duality theory for minimizing sums of the largest eigenvalues of symmetric matrices
    • M. L. Overton and R. S. Womersley, Optimality conditions and duality theory for minimizing sums of the largest eigenvalues of symmetric matrices, Math. Programming, 62 (1993), pp. 321-357.
    • (1993) Math. Programming , vol.62 , pp. 321-357
    • Overton, M.L.1    Womersley, R.S.2
  • 41
    • 0003668665 scopus 로고
    • The Symmetric Eigenvalue Problem
    • Prentice-Hall, Englewood Cliffs, NJ
    • B. N. Parlett, The Symmetric Eigenvalue Problem, Prentice-Hall, Englewood Cliffs, NJ, 1980.
    • (1980)
    • Parlett, B.N.1
  • 42
    • 11344286377 scopus 로고    scopus 로고
    • Fastest Mixing Markov Chain on Graphs with Symmetries
    • manuscript in preparation
    • P. A. Parrilo, L. Xiao, S. Boyd, and P. Diaconis, Fastest Mixing Markov Chain on Graphs with Symmetries, manuscript in preparation, 2004.
    • (2004)
    • Parrilo, P.A.1    Xiao, L.2    Boyd, S.3    Diaconis, P.4
  • 43
    • 0004267646 scopus 로고
    • Convex Analysis
    • Princeton University Press, Princeton, NJ
    • R. T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, NJ, 1970.
    • (1970)
    • Rockafellar, R.T.1
  • 44
    • 0000926886 scopus 로고
    • Convergence rates for Markov chains
    • J. S. Rosenthal, Convergence rates for Markov chains, Siam Rev., 37 (1995), pp. 387-405.
    • (1995) Siam Rev. , vol.37 , pp. 387-405
    • Rosenthal, J.S.1
  • 45
    • 0003521777 scopus 로고
    • Numerical Methods for Large Eigenvalue Problems
    • Manchester University Press, Manchester, UK
    • Y. Saad, Numerical Methods for Large Eigenvalue Problems, Manchester University Press, Manchester, UK, 1992.
    • (1992)
    • Saad, Y.1
  • 46
    • 32044432496 scopus 로고    scopus 로고
    • Semidefinite programming page
    • Website maintained by C. Helmberg
    • Semidefinite programming page, http://www-user.tu-chemnitz.de/ ~helmberg/semidef.html. Website maintained by C. Helmberg.
  • 47
    • 0003995427 scopus 로고
    • Minimization Methods for Non-differentiable Functions
    • Springer-Verlag, Berlin
    • N. Z. Shor, Minimization Methods for Non-differentiable Functions, Springer Ser. Comput. Math., Springer-Verlag, Berlin, 1985.
    • (1985) Springer Ser. Comput. Math.
    • Shor, N.Z.1
  • 48
    • 84971804758 scopus 로고
    • Improved bounds for mixing rates of Markov chains and multicommodity flow
    • A. Sinclair, Improved bounds for mixing rates of Markov chains and multicommodity flow, Combin. Probab. Comput., 1 (1992), pp. 351-370.
    • (1992) Combin. Probab. Comput. , vol.1 , pp. 351-370
    • Sinclair, A.1
  • 49
    • 0003445019 scopus 로고
    • Algorithms for Random Generation and Counting: A Markov Chain Approach
    • Birkhäuser Boston, Boston
    • A. Sinclair, Algorithms for Random Generation and Counting: A Markov Chain Approach, Birkhäuser Boston, Boston, 1993.
    • (1993)
    • Sinclair, A.1
  • 50
    • 0002236462 scopus 로고
    • Logarithmic Sobolev inequalities for Gibbs states
    • Lecture Notes in Math. 1563, Springer-Verlag, Berlin
    • D. Stroock, Logarithmic Sobolev inequalities for Gibbs states, in Dirichlet Forms (Varenna, 1992), Lecture Notes in Math. 1563, Springer-Verlag, Berlin, 1993, pp. 194-228.
    • (1993) Dirichlet Forms (Varenna, 1992) , pp. 194-228
    • Stroock, D.1
  • 51
    • 0033296299 scopus 로고    scopus 로고
    • Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones
    • 11-12, special issue on Interior Point Methods (CD supplement with software)
    • J. F. Sturm, Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones, Optim. Methods Softw., 11-12 (1999), pp. 625-653; special issue on Interior Point Methods (CD supplement with software).
    • Optim. Methods Softw. 1999 , pp. 625-653
    • Sturm, J.F.1
  • 52
    • 0000920501 scopus 로고    scopus 로고
    • Semidefinite optimization
    • M. Todd, Semidefinite optimization, Acta Numer., 10 (2001), pp. 515-560.
    • (2001) Acta Numer. , vol.10 , pp. 515-560
    • Todd, M.1
  • 53
    • 0033295027 scopus 로고    scopus 로고
    • SDPT3: A Matlab software package for semidefinite programming, version 2.1
    • K. C. Toh, M. J. Todd, and R. H. Tutuncu, SDPT3: A Matlab software package for semidefinite programming, version 2.1, Optim. Methods Softw., 11 (1999), pp. 545-581.
    • (1999) Optim. Methods Softw. , vol.11 , pp. 545-581
    • Toh, K.C.1    Todd, M.J.2    Tutuncu, R.H.3
  • 54
    • 0030106462 scopus 로고    scopus 로고
    • Semidefinite programming
    • L. Vandenberghe and S. Boyd, Semidefinite programming, Siam Rev., 38 (1996), pp. 49-95.
    • (1996) Siam Rev. , vol.38 , pp. 49-95
    • Vandenberghe, L.1    Boyd, S.2
  • 56
    • 0003857247 scopus 로고    scopus 로고
    • SDPSOL: A Parser/Solver for Semidefinite Programming and Determinant Maximization Problems with Matrix Structure. User's Guide Version Beta
    • Stanford University, Stanford, CA
    • S.-P. Wu and S. Boyd, SDPSOL: A Parser/Solver for Semidefinite Programming and Determinant Maximization Problems with Matrix Structure. User's Guide, Version Beta, Stanford University, Stanford, CA, 1996.
    • (1996)
    • Wu, S.-P.1    Boyd, S.2
  • 57
    • 0004212716 scopus 로고    scopus 로고
    • Interior Point Algorithms: Theory and Analysis
    • Wiley-Interscience Series in Discrete Mathematics and Optimization, John Wiley & Sons, New York
    • Y. Ye, Interior Point Algorithms: Theory and Analysis, Wiley-Interscience Series in Discrete Mathematics and Optimization, John Wiley & Sons, New York, 1997.
    • (1997)
    • Ye, Y.1


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