Iterative refinement for linear programming

INFORMS Journal on Computing

Volumn 28, Issue 3, 2016, Pages 449-464

  Gleixner, Ambros M ^a   Steffy, Daniel E ^b   Wolter, Kati ^c  

^a ZUSE INSTITUTE BERLIN   (Germany)

^b OAKLAND UNIVERSITY   (United States)

^c Mosek Aps   (Denmark)

Author keywords

Accurate solutions; Exact computation; High precision optimization; Iterative refinement; Linear programming; Rational arithmetic

Indexed keywords

ITERATIVE METHODS; OPTIMIZATION;

EXACT COMPUTATIONS; EXACT REPRESENTATIONS; HIGH-PRECISION; ITERATIVE REFINEMENT; NUMERIC COMPUTATION; PRECISION ARITHMETIC; PROBLEM DESCRIPTION; RATIONAL ARITHMETICS;

LINEAR PROGRAMMING;

EID: 84977641726     PISSN: 10919856     EISSN: 15265528     Source Type: Journal    
DOI: 10.1287/ijoc.2016.0692     Document Type: Article

References (48)

1. Althaus E, Dumitriu D (2009) Fast and accurate bounds on linear programs. Vahrenhold J, ed., Proc. 8th Internat. Sympos. Experiment. Algorithms, Lecture Notes in Computer Science, Vol. 5526 (Springer, Berlin), 40-50.
2. Applegate DL, Cook W, Dash S, Espinoza DG (2007a) Exact solutions to linear programming problems. Oper. Res. Lett. 35(6):693-699.
3. Applegate DL, Cook W, Dash S, Espinoza DG (2007b) QSopt-ex. Accessed April 19, 2016, http://www.dii.uchile.cl/~daespino/ESolver-doc/.
4. Avis D, Fukuda K (1992) A pivoting algorithm for convex hulls and vertex enumeration of arrangements and polyhedra. Discrete Comput. Geometry 8(1):295-313.
5. Azulay D-O, Pique J-F (1998) Optimized Q-pivot for exact linear solvers. Maher M, Puget J-F, eds. Principles and Practice of Constraint Programming CP98, Lecture Notes in Computer Science, Vol. 1520 (Springer, Berlin), 55-71.
6. Buchheim C, Chimani M, Ebner D, Gutwenger C, Jünger M, Klau G, Mutzel P, Weiskircher R (2008) A branch-and-cut approach to the crossing number problem. Discrete Optim. 5(2):373-388.
7. Bulutoglu DA, Kaziska DM (2010) Improved WLP and GWP lower bounds based on exact integer programming. J. Statist. Planning Inference 140(5):1154-1161.
8. Burton BA, Ozlen M (2012) Computing the crosscap number of a knot using integer programming and normal surfaces. ACM Trans. Math. Software 39(1):4:1-4:18.
9. Chindelevitch L, Trigg J, Regev A, Berger B (2014) An exact arithmetic toolbox for a consistent and reproducible structural analysis of metabolic network models. Nature Comm. 5:Article no. 4893.
10. Chvátal V (1983) Linear Programming (W. H. Freeman and Company, New York).
11. Cohn H, Jiao Y, Kumar A, Torquato S (2011) Rigidity of spherical codes. Geometry Topology 15(4):2235-2273.
12. Cook W, Steffy DE (2011) Solving very sparse rational systems of equations. ACM Trans. Math. Software 37(4):39:1-39:21.
13. Cook W, Koch T, Steffy DE, Wolter K (2011) An exact rational mixed-integer programming solver. Günlük O, Woeginger G, eds. Integer Programming and Combinatoral Optimization, Lecture Notes in Computer Science, Vol. 6655 (Springer, Berlin), 104-116.
14. Dantzig GB (1963) Linear Programming and Extensions (Princeton University Press, Princeton, NJ).
15. de Oliveira Filho FM, Vallentin F (2010) Fourier analysis, linear programming, and densities of distance avoiding sets in ℝn. J. Eur. Math. Soc. 12(6):1417-1428.
16. Dhiflaoui M, Funke S, Kwappik C, Mehlhorn K, Seel M, Schömer E, Schulte R, Weber D (2003) Certifying and repairing solutions to large LPs: How good are LP-solvers? Proc. 14th Ann. Sympos. Discrete Algorithms, SODA '03 (SIAM, Philadelphia), 255-256.
17. Edmonds J (1994) Exact pivoting. Presentation, ECCO VII, May 26-28, Milan, Italy.
18. Edmonds J, Maurras J-F (1997) Note sur les Q-matrices d'Edmonds. RAIRO. Recherche Opérationnelle 31(2):203-209.
19. Espinoza DG (2006) On linear programming, integer programming and cutting planes. Unpublished doctoral dissertation, Georgia Institute of Technology, Athens, GA.
20. Fukuda K, Prodon A (1996) Double description method revisited. Deza M, Euler R, Manoussakis I, eds. Combinatorics and Computer Science, Lecture Notes in Computer Science, Vol. 1120 (Springer, Berlin), 91-111.
21. Gärtner B (1999) Exact arithmetic at low cost - A case study in linear programming. Comput. Geometry 13(2):121-139.
22. Gleixner AM, Steffy DE, Wolter K (2012) Improving the accuracy of linear programming solvers with iterative refinement. Proc. 37th Internat. Sympos. Symbolic Algebraic Comput. (ACM, New York), 187-194.
23. Golub G, van Loan C (1983) Matrix Computations (Johns Hopkins University Press, Baltimore).
24. Grötschel M, Lovasz L, Schrijver A (1988) Geometric Algorithms and Combinatorial Optimization (Springer-Verlag, Berlin).
25. Hales TC (2005) A proof of the Kepler conjecture. Ann. Math. 162(3):1065-1185.
26. Held S, Cook W, Sewell E (2012) Maximum-weight stable sets and safe lower bounds for graph coloring. Math. Programming Comput. 4(4):363-381.
27. Hicks I, McMurray N (2007) The branchwidth of graphs and their cycle matroids. J. Combinatorial Theory Ser. B 97(5):681-692.
28. Jansson C (2004) Rigorous lower and upper bounds in linear programming. SIAM J. Optim. 14(3):914-935.
29. Khachiyan LG (1979) A polynomial algorithm in linear programming. Soviet Mathematics Doklady 20(1):191-194. [Translation].
30. Klotz E (2014) Identification, assessment and correction of illconditioning and numerical instability in linear and integer programs. Newman A, Leung J, eds. TutORials in Operations Research: Bridging Data and Decisions (INFORMS, Catonsville, MD), 54-108.
31. Koch T (2004) The final NETLIB-LP results. Oper. Res. Lett. 32(2): 138-142.
32. Ladányi L (2011) Personal communication, November 26. IBM T.J. Watson Research Center, Yorktown Heights, NY.
33. Lerman JA, Hyduke DR, Latif H, Portnoy VA, Lewis NE, Orth JD, Schrimpe-Rutledge AC, et al. (2012) In silico method for modelling metabolism and gene product expression at genome scale. Nature Comm. 3:Article no. 929.
34. Maes C (2013) Personal communication, September 6. Gurobi Optimization, Inc., Houston, TX.
35. Moore RE, Kearfott RB, Cloud MJ (2009) Introduction to Interval Analysis (Society for Industrial and Applied Mathematics, Philadelphia).
36. Neumaier A, Shcherbina O (2004) Safe bounds in linear and mixedinteger linear programming. Math. Programming 99(2):283-296.
37. Pan VY (2011) Nearly optimal solution of rational linear systems of equations with symbolic lifting and numerical initialization. Comput. Math. Appl. 62(4):1685-1706.
38. Renegar J (1994) Some perturbation theory for linear programming. Math. Programming 65(1):73-91.
39. Saunders BD, Wood DH, Youse BS (2011) Numeric-symbolic exact rational linear system solver. Proc. 36th Internat. Sympos. Symbolic Algebraic Comput., ISSAC '11 (ACM, New York), 305-312.
40. Saunders MA, Tenenblat L (2006) The zoom strategy for accelerating and warm-starting interior methods. Presentation, INFORMS Annual Meeting, Pittsburgh.
41. Schrijver A (1986) Theory of Linear and Integer Programming (Wiley, Chichester, UK).
42. Steffy DE, Wolter K (2013) Valid linear programming bounds for exact mixed-integer programming. INFORMS J. Comput. 25(2): 271-284.
43. Ursic S, Patarra C (1983) Exact solution of systems of linear equations with iterative methods. SIAM J. Matrix Anal. Appl. 4(1):111-115.
44. Von zur Gathen J, Gerhard J (2003) Modern Computer Algebra (Cambridge University Press, Cambridge, UK).
45. Wan Z (2006) An algorithm to solve integer linear systems exactly using numerical methods. J. Symbolic Comput. 41(6):621-632.
46. Wilkinson JH (1963) Rounding Errors in Algebraic Processes (Prentice Hall, Englewood Cliffs, NJ).
47. Wunderling R (1996) Paralleler und objektorientierter simplexalgorithmus. Unpublished doctoral thesis, Technische Universität Berlin.
48. Yap CK (1997) Robust geometric computation. Goodman JE, O'Rourke J, eds. Handbook of Discrete and Computational Geometry (CRC Press, Boca Raton, FL), 653-668.