Mathematical Programming Computation

, Volume 2, Issue 2, pp 103–124 | Cite as

Efficient high-precision matrix algebra on parallel architectures for nonlinear combinatorial optimization

Full Length Paper

Abstract

We provide a first demonstration of the idea that matrix-based algorithms for nonlinear combinatorial optimization problems can be efficiently implemented. Such algorithms were mainly conceived by theoretical computer scientists for proving efficiency. We are able to demonstrate the practicality of our approach by developing an implementation on a massively parallel architecture, and exploiting scalable and efficient parallel implementations of algorithms for ultra high-precision linear algebra. Additionally, we have delineated and implemented the necessary algorithmic and coding changes required in order to address problems several orders of magnitude larger, dealing with the limits of scalability from memory footprint, computational efficiency, reliability, and interconnect perspectives.

Keywords

Nonlinear combinatorial optimization Matroid optimization High-performance computing High-precision linear algebra 

Mathematics Subject Classification (2000)

90-08 90C27 90C26 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Alam, S., Barrett, R., Bast, M., Fahey, M.R., Kuehn, J., McCurdy, C., Rogers, J., Roth, P., Sankaran, R., Vetter, J.S., Worley, P., Yu, W.: Early evaluation of IBM BlueGene/P, SC ’08. In: Proceedings of the 2008 ACM/IEEE Conference on Supercomputing, pp 1–12 (2008)Google Scholar
  2. 2.
    Almási G., Archer C., Castaños J.G., Gunnels J.A., Erway C.C., Heidelberger P., Martorell X., Moreira J.E., Pinnow K., Ratterman J., Steinmacher-Burow B.D., Gropp W., Toonen B.: Design and implementation of message-passing services for the Blue Gene/L supercomputer. IBM. J. Res. Dev. 49(2–3), 393–406 (2005)CrossRefGoogle Scholar
  3. 3.
  4. 4.
    Bailey D.H.: High-precision arithmetic in scientific computation. Comput. Sci. Eng. 7(3), 54–61 (2005)CrossRefGoogle Scholar
  5. 5.
    Bailey, D.H., Borwein, J.M.: Highly Parallel, High-Precision Numerical Integration. LBNL-57491 (2005)Google Scholar
  6. 6.
    Berstein Y., Lee J., Maruri-Aguilar H., Onn S., Riccomagno E., Weismantel R., Wynn H.: Nonlinear matroid optimization and experimental design. SIAM J. Discret. Math. 22(3), 901–919 (2008)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Berstein, Y., Lee, J., Onn, S., Weismantel, R.: Parametric nonlinear discrete optimization over well-described sets and matroid intersections. Math. Program. (to appear)Google Scholar
  8. 8.
    Buttari A., Dongarra J., Langou J., Langou J., Luszczek P., Kurzak J.: Mixed precision iterative refinement techniques for the solution of dense linear systems. Int. J. High. Perform. Comput. Appl. 21(4), 457–466 (2007)CrossRefGoogle Scholar
  9. 9.
    Chiu, G.L.-T., Gupta, M., Royyuru, A.K. (guest eds.): Blue Gene. IBM J. Res. Dev. 49(2/3) (2005)Google Scholar
  10. 10.
    Choi J., Dongarra J., Ostrouchov S., Petitet A., Walker D., Whaley C.: Design and implementation of the ScaLAPACK LU, QR, and Cholesky factorization routines. Sci. Program. 5(3), 173–184 (1996)Google Scholar
  11. 11.
    De Loera, J., Haws, D.C., Lee, J., O’Hair, A.: Computation in multicriteria matroid optimization. ACM J. Exp. Algorithmics 14, Article No. 8, (2009)Google Scholar
  12. 12.
    Demmel, J.: The future of LAPACK and ScaLAPACK, PARA 06: workshop on State-of-the-Art. In: Scientific and Parallel Computing, Umea, Sweden, 18–21 June 2006. http://www.cs.berkeley.edu/~demmel/cs267_Spr07/future_sca-lapack_CS267_Spr07.ppt
  13. 13.
    Eisinberg A., Fedele G., Imbrogno C.: Vandermonde systems on equidistant nodes in [0, 1]: accurate computation. Appl. Math. Comput. 172, 971–984 (2006)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Eisinberg A., Franzé G., Pugliese P.: Vandermonde matrices on integer nodes. Numerische Mathematik 80(1), 75–85 (1998)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Fries A., Hunter W.G.: Minimum aberration 2k-p designs. Technometrics 22, 601–608 (1980)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Harvey, N.: Algebraic algorithms for matching and matroid problems. SIAM J. Comput. (2010, to appear)Google Scholar
  17. 17.
    Higham N.J.: Accuracy and Stability of Numerical Algorithms, 2nd edn. Society for Industrial and Applied Mathematics, Philadelphia, PA, USA (2002)MATHGoogle Scholar
  18. 18.
    IBM Blue Gene Team: Overview of the IBM Blue Gene/P project, IBM J. Res. Dev., v52, (2008), 199–220Google Scholar
  19. 19.
    Lee, J.: A First Course in Combinatorial Optimization. Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge (2004)Google Scholar
  20. 20.
    Lee J., Ryan J.: Matroid applications and algorithms. INFORMS (formerly ORSA) J. Comput. 4, 70–98 (1992)MATHMathSciNetGoogle Scholar
  21. 21.
  22. 22.
    Mulmuley K., Vazirani U.V., Vazirani V.V.: Matching is as easy as matrix inversion. Combinatorica 7, 105–113 (1987)MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Oxley J.G.: Matroid Theory. Oxford Science Publications, The Clarendon Press, Oxford University Press, New York (1992)MATHGoogle Scholar
  24. 24.
    Pistone, G., Riccomagno, E., Wynn, H.P.: Algebraic Statistics. Monographs on Statistics and Applied Probability, vol. 89. Chapman & Hall/CRC, Boca Raton (2001)Google Scholar
  25. 25.
    Recski A.: Matroid theory and its applications in electric network theory and in statics. Springer, Berlin (1989)Google Scholar
  26. 26.
    Tweddle, I.: James Stirling’s methodus differentialis: an annotated translation of Stirling’s text. In: Sources and Studies in the History of Mathematics and Physical Sciences, Springer (2003)Google Scholar
  27. 27.
    van de Geijn R.A., Watts J.: SUMMA: scalable universal matrix multiplication algorithm. Concurr. Pract. Experience 9(4), 255–274 (1997)CrossRefGoogle Scholar
  28. 28.
  29. 29.
    Wu H., Wu C.F.J.: Clear two-factor interactions and minimum aberration. Ann. Stat. 30, 1496–1511 (2002)MATHCrossRefGoogle Scholar

Copyright information

© Springer and Mathematical Programming Society 2010

Authors and Affiliations

  1. 1.IBM T.J. Watson Research CenterYorktown HeightsUSA
  2. 2.Department of Computational and Applied MathematicsRice UniversityHoustonUSA

Personalised recommendations