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Solving large-scale semidefinite programs in parallel

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Abstract

We describe an approach to the parallel and distributed solution of large-scale, block structured semidefinite programs using the spectral bundle method. Various elements of this approach (such as data distribution, an implicitly restarted Lanczos method tailored to handle block diagonal structure, a mixed polyhedral-semidefinite subdifferential model, and other aspects related to parallelism) are combined in an implementation called LAMBDA, which delivers faster solution times than previously possible, and acceptable parallel scalability on sufficiently large problems.

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References

  1. Benson S.J., Ye Y., Zhang X. (2000) Solving large-scale sparse semidefinite programs for combinatorial optimization. SIAM J. Optim. 10(2): 443–461

    Article  MathSciNet  Google Scholar 

  2. Burer S. (2003) Semidefinite programming in the space of partial positive semidefinite matrices. SIAM J. Optim. 14(1): 139–172

    Article  MathSciNet  Google Scholar 

  3. Burer, S., Monteiro, R.D.C.: A nonlinear programming algorithm for solving semidefinite programming via low-rank factorization. Math. Program. (Ser. B) 95, 329–357 (2003)

    Google Scholar 

  4. Burer, S., Monteiro, R.D.C.: Local minima and convergence in low-rank semidefinite programming. Math. Program. (2006)(to appear)

  5. Cullum J., Donath W., Wolfe P (1975). The minimization of certain nondifferentiable sums of eigenvalues of symmetric matrices. Math. Program. Study 3: 35–65

    MathSciNet  Google Scholar 

  6. Fletcher R. (1985) Semidefinite matrix constraints in optimization. SIAM J. Control Optim. 23: 493–523

    Article  MathSciNet  Google Scholar 

  7. Fujisawa, K., Kojima, M., Nakata, K., Yamashita, M.: SDPA User’s Manual – Version 6.00. Department of Mathematical and Computing Sciences, Tokyo Institute of Technology (2002)

  8. Fukuda M., Kojima M., Murota K., Nakata K. (2000) Exploiting sparsity in semidefinite programming via matrix completion I: general framework. SIAM J. Optim. 11(3): 647–674

    Article  MathSciNet  Google Scholar 

  9. Fukuda, M., Braams, B.J., Nakata, M., Overton, M.L., Percus, J.K., Yamashita, M., Zhao, Z.: Large-scale semidefinite programs in electronic structure calculations. Technical report B-413, Department of Mathematical and Computing Sciences, Tokyo Institute of Technology (2005)

  10. Helmberg, C.: SBmethod: a C++ implementation of the spectral bundle method. Technical report ZR-00-35, Konrad-Zuse-Zentrum für Informationstechnik, Berlin (2000)

  11. Helmberg, C.: Semidefinite programming for combinatorial optimization. Technical report ZR-00-34, TU Berlin, Konrad-Zuse-Zentrum, Berlin (2000)

  12. Helmberg C. (2003) Numerical evaluation of SBmethod. Math. Program. 95(2): 381–406

    Article  MathSciNet  Google Scholar 

  13. Helmberg C., Kiwiel K.C. (2002) A spectral bundle method with bounds. Math. Program. 93(2): 173–194

    Article  MathSciNet  Google Scholar 

  14. Helmberg C., Rendl F. (1999) A spectral bundle method for semidefinite programming. SIAM J. Optim. 10(3): 673–696

    Article  MathSciNet  Google Scholar 

  15. Hiriart–Urruty J–B., Lemaréchal C. (1993) Convex Analysis and Minimization Algorithms, vol. I & II. Springer, Berlin Heidelberg New York

    Google Scholar 

  16. Hiriart–Urruty J–B., Ye D. (1995) Sensitivity analysis of all eigenvalues of a symmetric matrix. Numer. Math. 70: 45–72

    Article  MathSciNet  Google Scholar 

  17. Kiwiel K.C. (1983) An aggregate subgradient method for nonsmooth convex minimization. Math. Program. 27: 320–341

    MathSciNet  Google Scholar 

  18. Kiwiel K.C. (1990) Proximity control in bundle methods for convex nondifferentiable minimization. Math. Program. 46:105–122

    Article  MathSciNet  Google Scholar 

  19. Kočvara, M., Stingl, M.: On the solution of large-scale SDP problems by the modified barrier method using iterative solvers. Technical report 304, Institute of Applied Mathematics, University of Erlangen (2005)

  20. Lehoucq R.B., Sorensen D.C., Yang C. (1998) ARPACK Users’ Guide. SIAM, Philadelphia

    Google Scholar 

  21. Mazziotti D.A. (2004) First-order semidefinite programming for the direct determination of two-electron reduced density matrices with application to many-electron atoms and molecules. J. Chem. Phys. 121: 10957–10966

    Article  Google Scholar 

  22. Mazziotti D.A. (2004) Realization of quantum chemistry without wavefunctions through first-order semidefinite programming. Phys. Rev. Lett. 93: 213001

    Article  Google Scholar 

  23. Mittelmann H.D. (2003) An independent benchmarking of SDP and SOCP solvers. Math. Program. 95: 407–430

    Article  MathSciNet  Google Scholar 

  24. Nakata K., Fujisawa K., Fukuda M., Kojima M., Murota K. (2003) Exploiting sparsity in semidefinite programming via matrix completion II: implementation and numerical results. Math. Program. 95(2): 303–327

    Article  MathSciNet  Google Scholar 

  25. Nayakkankuppam, M.V.: Optimization over symmetric cones. Ph.D. Thesis, New York University (1999)

  26. Nayakkankuppam, M.V., Tymofyeyev, Y.: A parallel implementation of the spectral bundle method for semidefinite programming. In: Proceedings of the 8th SIAM Conference on Applied Linear Algebra. SIAM, Williamsburg (2003)

  27. Overton M.L. (1988) On minimizing the maximum eigenvalue of a symmetric matrix. SIAM J. Matrix Anal. Appl. 9(2): 256–268

    Article  MathSciNet  Google Scholar 

  28. Parlett B.M. (1998) The Symmetric Eigenvalue Problem. SIAM, Philadelphia

    MATH  Google Scholar 

  29. Rockafellar R.T. (1970) Convex Analysis. Princeton University Press, Princeton

    MATH  Google Scholar 

  30. Sorensen D.C. (1992) Implicit application of polynomial filters in a k-step Arnoldi method. SIAM J. Sci. Comput. 13(1): 357–385

    MathSciNet  Google Scholar 

  31. Toh K.C. (2004) Solving large scale semidefinite programs via an iterative solver on the augmented systems. SIAM J. Optim. 14(3): 670–698

    Article  MathSciNet  Google Scholar 

  32. Toh K.C., Kojima M. (2002) Solving some large scale semidefinite programs via the conjugate residual method. SIAM J. Optim. 12(3): 669–691

    Article  MathSciNet  Google Scholar 

  33. Yamashita, M., Fujisawa, K., Kojima, M.: SDPARA: Semi Definite Programming Algorithm: paRAllel version. Parallel Comput. 29, 1053–1067 (2003). http://grid.r.dendai.ac.jp/sdpa

  34. Zhao Z., Braams B.J., Fukuda M., Overton M.L., Percus J.K. (2004) The reduced density matrix method for electronic structure calculations and the role of three-index representability conditions. J. Chem. Phys 120(5): 2095–2104

    Article  Google Scholar 

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Authors and Affiliations

  1. Computational Finance Group, Bloomberg LP, 731 Lexington Avenue, New York, NY, 10022, USA

    Madhu V. Nayakkankuppam

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  1. Madhu V. Nayakkankuppam
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Correspondence to Madhu V. Nayakkankuppam.

Additional information

Dedicated to the memory of Jos Sturm.

This work was supported in part by NSF grants DMS-0215373 and DMS-0238008.

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Nayakkankuppam, M.V. Solving large-scale semidefinite programs in parallel. Math. Program. 109, 477–504 (2007). https://doi.org/10.1007/s10107-006-0032-1

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  • DOI: https://doi.org/10.1007/s10107-006-0032-1

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