The double pivot simplex method

Fabio Vitor1, Todd Easton1
1Dept. of Industrial and Manufacturing Systems Eng., Kansas State University, Manhattan, USA

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Alterovitz R, Lessard E, Pouliot J, Hsu I, O’Brien J, Goldberg K (2006) Optimization of HDR brachytherapy dose distributions using linear programming with penalty costs. Med Phys 33(11):4012–4019

Appelgren L (1969) A column generation algorithm for a ship scheduling problem. Transp Sci 3(1):53–68

Bartels R (1971) A stabilization of the simplex method. Numer Math 16(5):414–434

Bartolini F, Bazzani G, Gallerani V, Raggi M, Viaggi D (2007) The impact of water and agriculture policy scenarios on irrigated farming systems in Italy: an analysis based on farm level multi-attribute linear programming models. Agric Syst 93(1):90–114

Bazaraa M, Jarvis J, Sherali H (2009) Linear programming and network flows. Wiley, New Jersey

Bertsimas D, Tsitsiklis J (1997) Introduction to linear optimization. Athena Scientific, Belmont

Bland R (1977) New finite pivoting rules for the simplex method. Math Oper Res 2(2):103–107

Chalermkraivuth K, Bollapragada S, Clark M, Deaton J, Kiaer L, Murdzek J, Neeves W, Scholz B, Toledano D (2005) GE asset management, Genworth financial, and GE insurance use a sequential-linear-programming algorithm to optimize portfolios. Interfaces 35(5):370–380

Coppersmith D, Winograd S (1990) Matrix multiplication via arithmetic progressions. J Symb Comput 9(3):251–280

Dantzig G (1947) Maximization of a linear function of variables subject to linear inequalities. In: Koopmans TC (ed) Activity analysis of production and allocation, 1951. Wiley, New York, pp 339–347

Dantzig G (1982) Reminiscences about the origins of linear programming. Oper Res Lett 1(2):43–48

Dantzig G, Orchard-Hays W (1954) The product form for the inverse in the simplex method. Math Tables Aids Comput 8(46):64–67

Dantzig G, Wolfe P (1960) Decomposition principle for linear programs. Oper Res 8(1):101–111

Dongarra J, Sullivan F (2000) Guest editors’ introduction: the top 10 algorithms. Comput Sci Eng 2(1):22–23

Dorfman R (1984) The discovery of linear programming. Ann Hist Comput 6(3):283–295

Dyer M (1984) Linear time algorithms for two- and three-variable linear programs. SIAM J Comput 13(1):31–45

Edmonds J (1967) Systems of distinct representatives and linear algebra. J Res Natl Bur Stand 71B(4):241–245

Eldersveld S, Saunders M (1992) A Block-LU update for large-scale linear programming. SIAM J Matrix Anal A 13(1):191–201

Elhallaoui I, Metrane A, Desaulniers G, Soumis F (2010) An improved primal simplex algorithm for degenerate linear programs. INFORMS J Comput 23(4):569–577

Ford L, Fulkerson D (1958) A suggested computation for maximal multi-commodity network flows. Manage Sci 5(1):97–101

Forrest J, Tomlin J (1972) Updated triangular factors of the basis to maintain sparsity in the product form simplex method. Math Program 2(1):263–278

García J, Florez J, Torralba A, Borrajo D, López C, García-Olaya Á, Sáenz J (2013) Combining linear programming and automated planning to solve intermodal transportation problems. Eur J Oper Res 227(1):216–226

Gass S, Vinjamuri S (2004) Cycling in linear programming problems. Comput Oper Res 31(2):303–311

Gautier A, Lamond B, Paré D, Rouleau F (2000) The québec ministry of natural resources uses linear programming to understand the wood-fiber market. Interfaces 30(6):32–48

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

Gilmore P, Gomory R (1961) A linear programming approach to the cutting-stock problem. Oper Res 9(6):849–859

Gilmore P, Gomory R (1963) A linear programming approach to the cutting-stock problem—part II. Oper Res 11(6):863–888

Goldfarb D, Todd M (1989) Linear programming. In: Nemhauser GL, Rinnooy Kan AHG, Todd MJ (eds) Handbooks in operations research and management science, vol 1. North-Holland, Amsterdam, pp 73–170

Gomes A, Oliveira J (2006) Solving irregular strip packing problems by hybridising simulated annealing and linear programming. Eur J Oper Res 171(3):811–829

Gondzio J (2012) Interior point methods 25 years later. Eur J Oper Res 218(3):587–601

He J (1999) Homotopy perturbation technique. Comput Methods Appl Mech Eng 178(3–4):257–262

Hillier F, Lieberman G (2015) Introduction to operations research. McGraw-Hill, New York

Howard R (1960) Dynamic programming and Markov processes. The MIT Press, Cambridge

Huangfu Q, Julian Hall J (2015) Novel update techniques for the revised simplex method. Comput Optim Appl 60(3):587–608

Illés T, Terlaky T (2002) Pivot versus interior point methods: pros and cons. Eur J Oper Res 140(2):170–190

Kantorovich L (1939) Mathematical methods of organizing and planning production. Manage Sci 6(4):366–422 (1939 Russian, 1960 English)

Karmarkar N (1984) A new polynomial-time algorithm for linear programming. Combinatorica 4(4):373–395

Khachiyan L (1979) A polynomial algorithm in linear programming. Sov Math Dokl 20(1):191–194

Klee V, Minty G (1972) How good is the simplex algorithm? In: Shisha O (ed) Inequalities-III: proceedings of the third symposium on inequalities. Academic Press, New York, pp 159–175

Koch T, Achterberg T, Andersen E, Bastert O, Berthold T, Bixby R, Danna E, Gamrath G, Gleixner A, Heinz S, Lodi A, Mittelmann H, Ralphs T, Salvagnin D, Steffy D, Wolter K (2011) MIPLIB 2010. Math Program Comput 3(2):103–163

Kojima M, Mizuno S, Yoshise A (1989) A primaldual interior point algorithm for linear programming. In: Megiddo N (ed) Progress in mathematical programming: interior-point algorithms and related methods. Springer, New York, pp 29–47

Kojima M, Megiddo N, Mizuno S (1993) A primal-dual infeasible-interior-point algorithm for linear programming. Math Program 61(1):263–280

Koopmans T (1949) Optimum utilization of the transportation system. Econometrica 17(Supplement):136–146

Kunnumkal S, Talluri K, Topaloglu H (2012) A randomized linear programming method for network revenue management with product-specific no-shows. Transport Sci 46(1):90–108

Lee E, Gallagher R, Patterson D (2003) A linear programming approach to discriminant analysis with a reserved-judgment region. INFORMS J Comput 15(1):23–41

Lustig IJ, Marsten RE, Shanno DF (1994) Interior point methods for linear programming: computational state of the art. ORSA J Comput 6(1):1–14

Mansini R, Ogryczak W, Speranza M (2007) Conditional value at risk and related linear programming models for portfolio optimization. Ann Oper Res 152(1):227–256

Megiddo N (1983) Linear-time algorithms for linear programming in $$\mathbb{R}^{3}$$ R 3 and related problems. SIAM J Comput 12(4):759–776

Megiddo N (1989) Pathways to the optimal set in linear programming. In: Megiddo N (ed) Progress in mathematical programming: interior-point algorithms and related methods. Springer, New York, pp 131–158

Mehrotra S (1992) On the implementation of a primal-dual interior point method. SIAM J Optim 2(4):575–601

Nadarajah S, Margot F, Secomandi N (2015) Relaxations of approximate linear programs for the real option management of commodity storage. Manage Sci 61(12):3054–3076

Padberg M (1999) Linear optimization and extensions. Algorithms and combinatorics, vol 12. Springer-Verlag

Press W, Teukolsky S, Vetterling W, Flannery B (2007) Numerical recipes. Cambridge University Press, New York

Raymond V, Soumis F, Orban D (2010) A new version of the improved primal simplex for degenerate linear programs. Comput Oper Res 37(1):91–98

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

Romeijn H, Ahuja R, Dempsey J, Kumar A (2006) A new linear programming approach to radiation therapy treatment planning problems. Oper Res 54(2):201–216

Rong A, Lahdelma R (2008) Fuzzy chance constrained linear programming model for optimizing the scrap charge in steel production. Eur J Oper Res 186(3):953–964

Schrijver A (1998) Theory of linear and integer programming. Wiley, New York

Shamos M, Hoey D (1976) Geometric intersection problems. In: Seventeenth annual IEEE symposium on foundations of computer science, pp 208–215

Spielman D, Teng S (2004) Smoothed analysis of algorithms: why the simplex algorithm usually takes polynomial time. J ACM 51(3):385–463

Spitter J, Hurkens C, de Kok A, Lenstra J, Negenman E (2005) Linear programming models with planned lead times for supply chain operations planning. Eur J Oper Res 163(3):706–720

Strassen V (1969) Gaussian elimination is not optimal. Numer Math 13(4):354–356

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

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

Tang L, Liu J, Rong A, Yang Z (2000) A mathematical programming model for scheduling steelmaking-continuous casting production. Eur J Oper Res 120(2):423–435

Terlaky T, Zhang S (1993) Pivot rules for linear programming: a survey on recent theoretical developments. Ann Oper Res 46(1):203–233

Todd M (1985) Linear and quadratic programming in oriented matroids. J Comb Theory 39(2):105–133

Tolla P (1986) A stable and sparsity exploiting LU factorization of the basis matrix in linear programming. Eur J Oper Res 24(2):247–251

Williams V (2012) An overview of the recent progress on matrix multiplication. ACM SIGACT News 34(3):57–69

Winston W (2004) Operations research: applications and algorithms. Duxbury Press, Belmont

Ye Y (2011) The simplex and policy-iteration methods are strongly polynomial for the markov decision problem with a fixed discount rate. Math Oper Res 36(4):593–603

Zhou P, Ang B (2008) Linear programming models for measuring economy-wide energy efficiency performance. Energy Policy 36(8):2911–2916