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Novel update techniques for the revised simplex method

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Abstract

This paper introduces three novel techniques for updating the invertible representation of the basis matrix when solving practical sparse linear programming problems using a high performance implementation of the dual revised simplex method, being of particular value when suboptimization is used. Two are variants of the product form update and the other permits multiple Forrest–Tomlin updates to be performed. Computational results show that one of the product form variants is significantly more efficient than the traditional approach, with its performance approaching that of the Forrest–Tomlin update for some problems. The other is less efficient, but valuable in the context of the dual revised simplex method with suboptimization. Results show that the multiple Forrest–Tomlin updates are performed with no loss of serial efficiency.

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References

  1. Hall, J.A.J., Huangfu, Q.: A high performance dual revised simplex solver. In: R. Wyrzykowski et al. (ed.) PPAM 2011, Part I. LNCS, vol. 7203, pp. 143–151. Springer, Heidelberg (2012)

  2. Lubin, M., Hall, J.A.J., Petra, C.G., Anitescu, M.: Parallel distributed-memory simplex for large-scale stochastic LP problems. Comput. Optim. Appl. 55(3), 571–596 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  3. Suhl, U.H., Suhl, L.M.: Computing sparse LU factorizations for large-scale linear programming bases. ORSA J. Comput. 2(4), 325–335 (1990)

    Article  MATH  Google Scholar 

  4. Tomlin, J.A.: Pivoting for size and sparsity in linear programming inversion routines. J. Inst. Math. Appl. 10, 289–295 (1972)

    Article  MATH  MathSciNet  Google Scholar 

  5. Dantzig, G.B., Orchard-Hays, W.: The product form for the inverse in the simplex method. Math. Comp. 8, 64–67 (1954)

    Article  MATH  MathSciNet  Google Scholar 

  6. Bartels, R.H.: A stabilization of the simplex method. Numer. Math. 16, 414–434 (1971)

    Article  MATH  MathSciNet  Google Scholar 

  7. Forrest, J.J.H., Tomlin, J.A.: Updated triangular factors of the basis to maintain sparsity in the product form simplex method. Math. Program. 2, 263–278 (1972)

    Article  MATH  MathSciNet  Google Scholar 

  8. Reid, J.K.: A sparsity-exploiting variant of the Bartels-Golub decomposition for linear programming bases. Math. Program. 24(1), 55–69 (1982)

    Article  MATH  Google Scholar 

  9. Suhl, U.H., Suhl, L.M.: A fast LU update for linear programming. Ann. Oper. Res. 43(1), 33–47 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  10. Eldersveld, S.K., Saunders, M.A.: A block-LU update for large-scale linear programming. SIAM J. Matrix Anal. Appl. 13, 191–201 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  11. Gill, P.E., Murray, W., Saunders, M.A., Wright, M.H.: Sparse matrix methods in optimization. SIAM J. Sci. Stat. Comput. 5, 562–589 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  12. Hall, J.A.J.: Sparse matrix algebra for active set methods in linear programming. Ph.D. thesis, University of Dundee Department of Mathematics and Computer Science (1991)

  13. Bisschop, J., Meeraus, A.J.: Matrix augmentation and partitioning in the updating of the basis inverse. Math. Program. 13, 241–254 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  14. Elble, J.M., Sahinidis, N.V.: A review of the LU update in the simplex algorithm. Int. J. Math. Oper. Res. 4(4), 366–399 (2012)

    Article  MathSciNet  Google Scholar 

  15. Huangfu, Q., Hall, J.A.J.: Parallel dual simplex methods. School of Mathematics, University of Edinburgh, Technical Report (2014). (In preparation)

  16. Hall, J.A.J., McKinnon, K.I.M.: Hyper-sparsity in the revised simplex method and how to exploit it. Comput. Optim. Appl. 32(3), 259–283 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  17. Tomlin, J.A.: On pricing and backward transformation in linear programming. Math. Program. 6, 42–47 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  18. Rosander, R.R.: Multiple pricing and suboptimization in dual linear programming algorithms. Math. Program. Study 4, 108–117 (1975)

    Article  MathSciNet  Google Scholar 

  19. Bartels, R.H., Golub, G.H.: The simplex method of linear programming using LU decomposition. Commun. ACM 12(5), 266–268 (1969)

    Article  MATH  Google Scholar 

  20. COIN-OR: Clp. http://www.coin-r.org/projects/Clp.xml. Accessed 26 June 2013

  21. Koberstein, A.: Progress in the dual simplex algorithm for solving large scale LP problems: techniques for a fast and stable implementation. Comput. Optim. Appl. 41(2), 185–204 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  22. Gilbert, J.R., Peierls, T.: Sparse partial pivoting in time proportional to arithmetic operations. SIAM J. Sci. Stat. Comput. 9(5), 862–874 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  23. Gay, D.M.: Electronic mail distribution of linear programming test problems. Math. Program. Soc. COAL Newslett. 13, 10–12 (1985)

    Google Scholar 

  24. Carolan, W.J., Hill, J.E., Kennington, J.L., Niemi, S., Wichmann, S.J.: An empirical evaluation of the KORBX algorithms for military airlift applications. Oper. Res. 38(2), 240–248 (1990)

    Article  Google Scholar 

  25. IBM: ILOG CPLEX Optimizer. http://www.ibm.com/software/products/gb/en/ibmilogcpleoptistud/. Accessed 26 June 2013

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Authors and Affiliations

  1. School of Mathematics, University of Edinburgh, JCMB, King’s Buildings, Edinburgh, EH9 3JZ, UK

    Qi Huangfu & J. A. Julian Hall

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  1. Qi Huangfu
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  2. J. A. Julian Hall
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Correspondence to J. A. Julian Hall.

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Huangfu, Q., Hall, J.A.J. Novel update techniques for the revised simplex method. Comput Optim Appl 60, 587–608 (2015). https://doi.org/10.1007/s10589-014-9689-1

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