Towards a practical parallelisation of the simplex method

Computational Management Science

Volumn 7, Issue 2, 2010, Pages 139-170

  Hall, J A J ^a  

^a UNIVERSITY OF EDINBURGH   (United Kingdom)

Author keywords

Linear programming; Parallel computing; Simplex method; Sparse

Indexed keywords

EID: 77954087804     PISSN: 1619697X     EISSN: 16196988     Source Type: Journal    
DOI: 10.1007/s10287-008-0080-5     Document Type: Article

References (76)

1. Agrawal A, Blelloch GE, Krawitz RL, Phillips CA (1989) Four vector-matrix primitives. In: ACM symposium on parallel algorithms and architectures, pp 292-302.
2. Amdahl GM (1967) Validity of the single-processor approach to achieving large scale computing capabilities. In: AFIPS conference proceedings, vol 30. AFIPS Press, Reston, pp 483-485.
3. Amestoy PR, Duff IS, L'Excellent J-Y (2000) Multifrontal parallel distributed symmetric and unsymmetric solvers. Comput Methods Appl Mech Eng 184: 501-520.
4. Amestoy PR, Duff IS, Koster J, L'Excellent J-Y (2001) A fully asynchronous multifrontal solver using distributed dynamic scheduling. SIAM J Matrix Anal Appl 23(1): 15-41.
5. Amestoy PR, Guermouche A, L'Excellent J-Y, Pralet S (2006) Hybrid scheduling for the parallel solution of linear systems. Parallel Comput 32(2): 136-156.
6. Amestoy P, Li XS, Ng EG (2007a) Diagonal Markowitz scheme with local symmetrization. SIAM J Matrix Anal Appl 29(1): 228-244.
7. Amestoy P, Pralet S, Li XS (2007b) Unsymmetric orderings using a constrained Markowitz scheme. SIAM J Matrix Anal Appl 29(1): 302-327.
8. Aykanat C, Pinar A, Çatalyürek ÜV (2004) Permuting sparse rectangular matrices into block-diagonal form. SIAM J Sci Comput 25(6): 1860-1879.
9. Babaev DA, Mardanov SS (1991) A parallel algorithm for solving linear programming problems. Zh Vychislitel'noi Matematiki Matematicheskoi Fiziki 31(1): 86-95.
10. Badr E-S, Moussa M, Papparrizos K, Samaras N, Sifaleras A (2006) Some computational results on MPI parallel implementations of dense simplex method. Trans Eng Comput Technol 17: 228-231.
11. Barr RS, Hickman BL (1994) Parallel simplex for large pure network problems: computational testing and sources of speedup. Oper Res 42(1): 65-80.
12. Beasley JE (1990) Linear programming on Cray supercomputers. J Oper Res Soc 41(2): 133-139.
13. Benders JF (1962) Partitioning procedures for solving mixed-variables programming problems. Numer Math 4: 238-252.
14. Bixby RE (2002) Solving real-world linear programs: a decade and more of progress. Oper Res 50(1): 3-15.
15. Bixby RE, Martin A (2000) Parallelizing the dual simplex method. INFORMS J Comput 12: 45-56.
16. Bixby RE, Fenelon M, Gu Z, Rothberg E, Wunderling R (2000) MIP: theory and practice closing the gap. In: Powell MJD, Scholtes S (eds) System modelling and optimization: methods, theory and applications. Kluwer, The Netherlands, pp 19-49.
17. Bodurog̃lu İİ (1997) Scalable massively parallel simplex algorithms for block-structured linear programs. Ph. D. thesis, GSAS, Columbia University, New York.
18. Boffey TB, Hay R (1989) Implementing parallel simplex algorithms. In: CONPAR 88. Cambridge, UK, Cambridge University Press, pp 169-176.
19. Carolan WJ, Hill JE, Kennington JL, Niemi S, Wichmann SJ (1990) An empirical evaluation of the KORBX algorithms for military airlift applications. Oper Res 38(2): 240-248.
20. Chang MD, Engquist M, Finkel R, Meyer RR (1988) A parallel algorithm for generalized networks. Ann Oper Res 14: 125-145.
21. Crowder H, Hattingh JM (1975) Partially normalized pivot selection in linear programming. Math Program Study 4: 12-25.
22. Cvetanovic Z, Freedman EG, Nofsinger C (1991) Efficient decomposition and performance of parallel PDE, FFT, Monte-Carlo simulations, simplex, and sparse solvers. J Supercomput 5: 19-38.
23. Dantzig GB (1960) The decomposition principle for linear programs. Oper Res 8: 101-111.
24. Dantzig GB, Orchard-Hays W (1954) The product form for the inverse in the simplex method. Math Comput 8: 64-67.
25. Demmel JW, Eisenstat SC, Gilbert JR, Li XS, Liu JWH (1999) A supernodal approach to sparse partial pivoting. SIAM J Matrix Anal Appl 20(3): 720-755.
26. Dempster MAH, Thompson RT (1998) Parallelization and aggregation of nested Benders decomposition. Ann Oper Res 81: 163-187.
27. Eckstein J, Bodurog̃lu İİ, Polymenakos L, Goldfarb D (1995) Data-parallel implementations of dense simplex methods on the connection machine CM-2. ORSA J Comput 7(4): 402-416.
28. Evans DJ, Hatzopoulos M (1979) A parallel linear system solver. Int J Comput Math 7: 227-238.
29. Ferris MC, Horn JD (1998) Partitioning mathematical programs for parallel solution. Math Program 80: 35-61.
30. Finkel RA (1987) Large-grain parallelism: three case studies. In: Jamieson LH, Gannon D, Douglas RJ (eds) The characteristics of parallel algorithms. MIT Press, Cambridge, pp 21-63.
31. Forrest JJ, Goldfarb D (1992) Steepest-edge simplex algorithms for linear programming. Math Program 57: 341-374.
32. Forrest JJH, Tomlin JA (1990) Vector processing in the simplex and interior methods for linear programming. Ann Oper Res 22: 71-100.
33. Gay DM (1985) Electronic mail distribution of linear programming test problems. Math Program Soc COAL Newsl 13: 10-12.
34. Gill PE, Murray W, Saunders MA, Wright MH (1989) A practical anti-cycling procedure for linearly constrained optimization. Math Program 45: 437-474.
35. Gnanendran SK, Ho JK (1993) Load balancing in the parallel optimization of block-angular linear programs. Math Program 62: 41-67.
36. Goldfarb D, Reid JK (1977) A practical steepest-edge simplex algorithm. Math Program 12: 361-371.
37. Hall JAJ (2005) SYNPLEX: a task-parallel scheme for the revised simplex method. In: Contributed talk at the second international workshop on combinatorial scientific computing (CSC05), June 2005.
38. Hall JAJ (2007) Parallel basis matrix triangularisation for hyper-sparse LP problems. In: Contributed talk the IMA conference on numerical linear algebra and optimisation, September 2007.
39. Hall JAJ, McKinnon KIM (1992) Update procedures for the parallel revised simplex method. Technical report MSR 92-13, Department of Mathematics and Statistics, University of Edinburgh.
40. Hall JAJ, McKinnon KIM (1996) PARSMI: a parallel revised simplex algorithm incorporating minor iterations and Devex pricing. In: Waśniewski J, Dongarra J, Madsen K, Olesen D (eds) Applied parallel computing, vol 1184. Lecture notes in computer science. Springer, Berlin, pp 67-76.
41. Hall JAJ, McKinnon KIM (1998) ASYNPLEX: an asynchronous parallel revised simplex method algorithm. Ann Oper Res 81: 27-49.
42. Hall JAJ, McKinnon KIM (1999) Exploiting hyper-sparsity in the revised simplex method. Technical report MS99-014. Department of Mathematics and Statistics, University of Edinburgh.
43. Hall JAJ, McKinnon KIM (2005) Hyper-sparsity in the revised simplex method and how to exploit it. Comput Optim Appl 32(3): 259-283.
44. Harris PMJ (1973) Pivot selection methods of the Devex LP code. Math Program 5: 1-28.
45. Helgason RV, Kennington LJ, Zaki HA (1988) A parallelisation of the simplex method. Ann Oper Res 14: 17-40.
46. Hiller RS, Eckstein J (1993) Stochastic dedication: designing fixed income portfolios using massively parallel Benders decomposition. Manage Sci 39(11): 1422-1438.
47. Ho JK, Lee TC, Sundarraj RP (1988) Decomposition of linear programs using parallel computation. Math Program 42: 391-405.
48. Ho JK, Sundarraj RP (1994) On the efficacy of distributed simplex algorithms for linear programming. Comput Optim Appl 3(4): 349-363.
49. Kaul RN (1965) An extension of generalized upper bounding techniques for linear programming. Technical Report ORC 65-27. In: Center OR, Berkley UC, San Fransisco, CA.
50. Klabjan D, Johnson EL, Nemhauser GL (2000) A parallel primal-dual simplex algorithm. Oper Res Lett 27: 47-55.
51. Kumar V, Grama A, Gupta A, Karypis G (2003) Introduction to parallel computing: design and analysis of algorithms, 2nd edn. Addison-Wesley, New York.
52. Lentini M, Reinoza A, Teruel A, Guillen A (1995) SIMPAR: a parallel sparse simplex. Comput Appl Math 14(1): 49-58.
53. Li XS, Demmel JW (2003) SuperLU_DIST: a scalable distributed-memory sparse direct solver for unsymmetric linear systems. ACM Trans Math Softw 29(2): 110-140.
54. Luo J, Reijns GL (1992) Linear programming on transputers. In: van Leeuwen J (eds) Algorithms, software, architecture IFIP. transactions A (computer science and technology). Elsevier, Amsterdam, pp 525-534.
55. Luo J, Reijns GL, Bruggeman F, Lindfield GR (1991) A survey of parallel algorithms for linear programming. In: Deprettere EF, van der Veen A-J (eds) Algorithms and parallel VLSI architectures, vol B. Elsevier, Amsterdam, pp 485-490.
56. Markowitz H (1957) The elimination form of the inverse and its application to linear programming. Manage Sci 3: 255-296.
57. Maros I, Mitra G (2000) Investigating the sparse simplex algorithm on a distributed memory multiprocessor. Parallel Comput 26: 151-170.
58. McKinnon KIM, Plab F (1997a) A modified Markowitz criterion to increase parallelism in inverse factors of sparse matrices. Technical report, Department of Mathematics and Statistics, University of Edinburgh.
59. McKinnon KIM, Plab F (1997b) An upper bound on parallelism in the forward transformation within the revised simplex method. Technical report, Department of Mathematics and Statistics, University of Edinburgh.
60. Medhi D (1990) Parallel bundle-based decomposition algorithm for large-scale structured mathematical programming problems. Ann Oper Res 22: 101-127.
61. Mittelmann HD (2008) Benchmarks for optimization software. http://plato.asu.edu/bench.html. Accessed February 2008.
62. Nielsen SN, Zenios SA (1997) Scalable Benders decomposition for stochastic linear programming. Parallel Comput 23: 1069-1088.
63. Orchard-Hays W (1968) Advanced linear programming computing techniques. McGraw-Hill, New York.
64. Peters J (1990) The network simplex method on a multiprocessor. Networks 20: 845-859.
65. Pfefferkorn CE, Tomlin JA (1976) Design of a linear programming system for the ILLIAC IV. Technical report SOL 76-8. Systems Optimization Laboratory, Stanford University.
66. Pinar A, Aykanat C (1996) An effective model to decompose linear programs for parallel solution. In: Waśniewski J, Dongarra J, Madsen K, Olesen D (eds) Applied parallel computing. Lecture notes in computer science, vol 1184. Springer, Heidelberg, pp 592-601.
67. Rosen JB, Maier RS (1990) Parallel solution of large-scale, block-angular linear programs. Ann Oper Res 22: 23-41.
68. Shu W (1995) Parallel implementation of a sparse simplex algorithm on MIMD distributed memory computers. J Parallel Distrib Comput 31(1): 25-40.
69. Stunkel CB (1988) Linear optimization via message-based parallel processing. In: International conference on parallel processing, vol III, August 1988, pp 264-271.
70. Suhl UH, Suhl LM (1990) Computing sparse LU factorizations for large-scale linear programming bases. ORSA J Comput 2(4): 325-335.
71. Thomadakis ME, Liu J-C (1996) An efficient steepest-edge simplex algorithm for SIMD computers. In: International conference on supercomputing, pp 286-293.
72. Tomlin JA (1972) Pivoting for size and sparsity in linear programming inversion routines. J Inst Maths Applics 10: 289-295.
73. Wunderling R (1996a) Parallelizing the simplex algorithm. ILAY workshop on linear algebra in optimzation, Albi.
74. Wunderling R (1996b) Paralleler und objektorientierter simplex. Technical report TR-96-09, Konrad-Zuse-Zentrum für Informationstechnik Berlin.
75. Yarmish G (2001) A distributed implementation of the simplex method. Ph. D. thesis, Polytechnic University, Brooklyn, NY, March 2001.
76. Zenios SA (1989) Parallel numerical optimization: current status and annotated bibliography. ORSA J Comput 1(1)(Winter): 20-43.