Skip to main content
SpringerLink
Log in
Book cover

Linear Programming Computation pp 441–460Cite as

Criss-Cross Simplex Method

  • Chapter
  • First Online:

  • 3049 Accesses

Abstract

Methods perform very differently when solving a same problem. It is common that a problem that is solved slowly by the simplex method would be solved fast by the dual simplex method; and vise versa. Consequently, LP packages often include multiple options to be chosen, as it seems to be impossible to predetermine which method would be better to solve a given problem. Pursuing a method with features of both the primal and dual methods, scholars have attempted to combine primal and dual simplex methods for years. A class of resulting variants may be described by “criss-cross” because of their switching between primal and dual iterations. The primal-dual algorithm (Sect. 7.1) and the self-dual parametric simplex algorithm (Sect. 7.2) may be also classified into this category.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   119.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

References

  • Abadie J, Corpentier J (1969) Generalization of the Wolfe reduced gradient method to the case of mon-linear constrained optimization. In: Fletcher R (ed) Optimization. Academic, London, pp 37–48

    Google Scholar 

  • Abel P (1987) On the choice of the pivot columns of the simplex method: gradient criteria. Computing 38:13–21

    MATH  MathSciNet  Google Scholar 

  • Adler I, Megiddo N (1985) A simplex algorithm whose average number of steps is bounded between two quadratic functions of the smaller dimension. J ACM 32:871–895

    MATH  MathSciNet  Google Scholar 

  • Adler I, Resende MGC, Veige G, Karmarkar N (1989) An implementation of Karmarkar’s algorithm for linear programming. Math Program 44:297–335

    MATH  Google Scholar 

  • Andersen E, Andersen K (1995) Presolving in linear programming. Math Program 71:221–245

    MATH  Google Scholar 

  • Andersen ED, Gondzio J, Mészáros C, Xu X (1996) Implementation of interior-point methods for large scale linear programming. In: Terlaky T (ed) Interior point methods of mathematical programming. Kluwer, Dordrecht

    Google Scholar 

  • Andrel N, Barbulescu M (1993) Balance constraints reduction of large-scale linear programming problems. Ann Oper Res 43:149–170

    MathSciNet  Google Scholar 

  • Anstreicher KM, Watteyne P (1993) A family of search directions for Karmarkar’s algorithm. Oper Res 41:759–767

    MATH  MathSciNet  Google Scholar 

  • Arrow KJ, Hurwicz L (1956) Reduction of constrained maxima to saddle-point problems. In: Neyman J (ed) Proceedings of the third Berkeley symposium on mathematical statistics and probability, vol 5. University of California Press, Berkeley, pp 1–26

    Google Scholar 

  • Avis D, Chvatal V (1978) Notes on Bland’s pivoting rule. Math Program 8:24–34

    Google Scholar 

  • Balas E (1965) An additive algorithm for solving linear programs with zero-one variables. Oper Res 13:517–546

    MathSciNet  Google Scholar 

  • Balinski ML, Gomory RE (1963) A mutual primal-dual simplex method. In: Graves RL, Wolfe P (eds) Recent advances in mathematical programming. McGraw-Hill, New York

    Google Scholar 

  • Balinski ML, Tucker AW (1969) Duality theory of linear problems: a constructive approach with applications. SIAM Rev 11:347–377

    MATH  MathSciNet  Google Scholar 

  • Barnes ER (1986) A variation on Karmarkars algorithm for solving linear programming problems. Math Program 36:174–182

    MATH  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  • Bartels RH, Golub GH (1969) The simplex method of linear programming using LU decomposition. Commun ACM 12:266–268

    MATH  Google Scholar 

  • Bartels RH, Stoer J, Zenger Ch (1971) A realization of the simplex method based on triangular decompositions. In: Wilkinson JH, Reinsch C (eds) Contributions I/II in handbook for automatic computation, volume II: linear algebra. Springer, Berlin/London

    Google Scholar 

  • Bazaraa MS, Jarvis JJ, Sherali HD (1977) Linear programming and network flows, 2nd edn. Wiley, New York

    MATH  Google Scholar 

  • Beale EML (1954) An alternative method for linear programming. Proc Camb Philos Soc 50:513–523

    MATH  MathSciNet  Google Scholar 

  • Beale EML (1955) Cycling in the dual simplex algorithm. Nav Res Logist Q 2:269–275

    MathSciNet  Google Scholar 

  • Beale E (1968) Mathematical programming in practice. Topics in operations research. Pitman & Sons, London

    Google Scholar 

  • Benders JF (1962) Partitioning procedures for solving mixed-variables programming problems. Numer Math 4:238–252

    MATH  MathSciNet  Google Scholar 

  • Benichou MJ, Cautier J, Hentges G, Ribiere G (1977) The efficient solution of large scale linear programming problems. Math Program 13:280–322

    MATH  Google Scholar 

  • Bixby RE (1992) Implementing the simplex method: the initial basis. ORSA J Comput 4:287–294

    MathSciNet  Google Scholar 

  • Bixby RE (1994) Progress in linear programming. ORSA J Comput 6:15–22

    MATH  MathSciNet  Google Scholar 

  • Bixby RE (2002) Solving real-world linear problems: a decade and more of progress. Oper Res l50:3–15

    Google Scholar 

  • Bixby RE, Saltzman MJ (1992) Recovering an optimal LP basis from the interior point solution. Technical report 607, Dapartment of Mathematical Sciences, Clemson University, Clemson

    Google Scholar 

  • Bixby RE, Wagner DK (1987) A note on detecting simple redundancies in linear systems. Oper Res Lett 6:15–17

    MATH  MathSciNet  Google Scholar 

  • Bixby RE, Gregory JW, Lustig IJ, Marsten RE, Shanno DF (1992) Very large-scale linear programming: a case study in combining interior point and simplex methods. Oper Res 40:885–897

    MATH  MathSciNet  Google Scholar 

  • Björck A, Plemmons RJ, Schneider H (1981) Large-scale matrix problems. North-Holland, Amsterdanm

    MATH  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  • Borgwardt K-H (1982a) Some distribution-dependent results about the asymptotic order of the arerage number of pivot steps of the simplex method. Math Oper Res 7:441–462.

    MATH  MathSciNet  Google Scholar 

  • Borgwardt K-H (1982b) The average number of pivot steps required by the simplex method is polynomial. Z Oper Res 26:157–177

    MATH  MathSciNet  Google Scholar 

  • Botsaris CA (1974) Differential gradient methods. J Math Anal Appl 63:177–198

    MathSciNet  Google Scholar 

  • Breadley A, Mitra G, Williams HB (1975) Analysis of mathematica problems prior to applying the simplex algorithm. Math Program 8:54–83

    Google Scholar 

  • Brown AA, Bartholomew-Biggs MC (1987) ODE vs SQP methods for constrained optimization. Technical report, 179, The Numerical Center, Hatfield Polytechnic

    Google Scholar 

  • Carolan MJ, Hill JE, Kennington JL, Niemi S, Wichmann SJ (1990) An empirical evaluation of the KORBX algorithms for military airlift applications. Oper Res 38:240–248

    Google Scholar 

  • Cavalier TM, Soyster AL (1985) Some computational experience and a modification of the Karmarkar algorithm. ISME working paper, The Pennsylvania State University, pp 85–105

    Google Scholar 

  • Chan TF (1985) On the existence and computation of LU factorizations with small pivots. Math Comput 42:535–548

    Google Scholar 

  • Chang YY (1979) Least index resolution of degeneracy in linear complementarity problems. Technical Report 79–14, Department of OR, Stanford University

    Google Scholar 

  • Charnes A (1952) Optimality and degeneracy in linear programming. Econometrica 20:160–170

    MATH  MathSciNet  Google Scholar 

  • Cheng MC (1987) General criteria for redundant and nonredundant linear inequalities. J Optim Theory Appl 53:37–42

    MATH  MathSciNet  Google Scholar 

  • Chvatal V (1983) Linear programming. W.H. Freeman, New York

    MATH  Google Scholar 

  • Cipra BA (2000) The best of the 20th century: editors name top 10 algorithms. SIAM News 33: 1–2

    Google Scholar 

  • Coleman TF, Pothen A (1987) The null space problem II. Algorithms. SIAM J Algebra Discret Methods 8:544–562

    MathSciNet  Google Scholar 

  • Cook AS (1971) The complexity of theorem-proving procedure. In: Proceedings of third annual ACM symposium on theory of computing, Shaker Heights, 1971. ACM, New York, pp 151–158

    Google Scholar 

  • Cottle RW, Johnson E, Wets R (2007) George B. Dantzig (1914–2005). Not AMS 54:344–362

    MATH  MathSciNet  Google Scholar 

  • CPLEX ILOG (2007) 11.0 User’s manual. ILOG SA, Gentilly, France

    Google Scholar 

  • Curtis A, Reid J (1972) On the automatic scaling of mtrices for Gaussian elimination. J Inst Math Appl 10:118–124

    MATH  Google Scholar 

  • Dantzig GB (1948) Programming in a linear structure, Comptroller, USAF, Washington, DC

    Google Scholar 

  • Dantzig GB (1951a) Programming of interdependent activities, mathematical model. In: Koopmas TC (ed) Activity analysis of production and allocation. Wiley, New York, pp 19–32; Econometrica 17(3/4):200–211 (1949)

    Google Scholar 

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

    Google Scholar 

  • Dantzig GB (1951c) A proof of the equivalence of the programming problem and the game problem. In: Koopmans T (ed) Activity analysis of production and allocation. Wiley, New York, pp 330–335

    Google Scholar 

  • Dantzig GB (1963) Linear programming and extensions. Princeton University Press, Princeton

    MATH  Google Scholar 

  • Dantzig GB (1991) linear programming. In: Lenstra JK, Rinnooy Kan AHG, Schrijver A (eds) History of mathematical programming. CWI, Amsterdam, pp 19–31

    Google Scholar 

  • Dantzig GB, Ford LR, Fulkerson DR (1956) A primal-dual algorithm for linear programs. In: Kuhn HK, Tucker AW (eds) Linear inequalities and related systems. Princeton University Press, Princeton, pp 171–181

    Google Scholar 

  • Dantzig GB, Orchard-Hays W (1953) Alternate algorithm for the revised simplex method using product form for the inverse. Notes on linear programming: part V, RM-1268. The RAND Corporation, Santa Monica

    Google Scholar 

  • Dantzig GB, Orchard-Hayes W (1954) The product form for the inverse in the simplex method. Math Tables Other Aids Comput 8:64–67

    MATH  Google Scholar 

  • Dantzig GB, Thapa MN (1997) Linear programming 1: introduction. Springer, New York

    MATH  Google Scholar 

  • Dantzig GB, Thapa MN (2003) Linear programming 2: theory and extensions. Springer, New York

    Google Scholar 

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

    MATH  Google Scholar 

  • Dantzig GB, Orden A, Wolfe P (1955) The generalized simplex method for minimizing a linear form under linear inequality constraints. Pac J Math 5:183–195

    MATH  MathSciNet  Google Scholar 

  • de Ghellinck G, Vial J-Ph, Polynomial (1986) Newton method for linear programming. Algorithnica 1:425–453. (Special issue)

    Google Scholar 

  • Dikin I (1967) Iterative solution of problems of linear and quadratic programming. Sov Math Dokl 8:674–675

    MATH  Google Scholar 

  • Dikin I (1974) On the speed of an iterative process. Upravlyaemye Sistemi 12:54–60

    Google Scholar 

  • Dorfman R, Samuelson PA, Solow RM (1958) Linear programming and economic analysis. McGraw-Hill, New York

    MATH  Google Scholar 

  • Duff IS, Erisman AM, Reid JK (1986) Direct methods for sparse matrices. Oxford University Press, Oxford

    MATH  Google Scholar 

  • Evtushenko YG (1974) Two numerical methods for solving non-linear programming problems. Sov Math Dokl 15:420–423

    MATH  Google Scholar 

  • Fang S-C (1993) Linear optimization and extensions: theory and algorithms. AT & T, Prentice-Hall, Englewood Cliffs

    MATH  Google Scholar 

  • Farkas J (1902) Uber die Theorie der Einfachen Ungleichungen. Journal für die Reine und Angewandte Mathematik 124:1–27

    Google Scholar 

  • Fiacco AV, Mccormick GP (1968) Nonlinear programming: sequential unconstrained minimization techniques. Wiley, New York

    MATH  Google Scholar 

  • Fletcher R (1981) Practical methods of optimization. Volume 2: constrained optimization. Wiley, Chichester

    Google Scholar 

  • Ford Jr LR, Fulkerson DR (1956) Maximal flow through a network. Can J Math 8:399–407

    MATH  MathSciNet  Google Scholar 

  • Forrest JJH, Goldfarb D (1992) Steepest edge simplex algorithm for linear programming. Math Program 57:341–374

    MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  • Forsythe GE, Malcom MA, Moler CB (1977) Computer methods for mathematical computations. Pretice-Hall, EnglewoodCliffs

    MATH  Google Scholar 

  • Fourer R (1979) Sparse Gaussian elimination of staircase linear systems. Tehchnical report ADA081856, Calif Systems Optimization LAB, Stanford University

    Google Scholar 

  • Fourier JBJ (1823) Analyse des travaux de l’Academie Royale des Science, pendant l’Iannee, Partie mathematique, Histoire de l’Acanemie Royale des Sciences de l’nstitut de France 6 [1823] (1826), xxix–xli (partially reprinted as: Premier extrait, in Oeuvres de Fourier, Tome 11 (Darboux G, ed.), Gauthier-Villars, Paris, 1890, (reprinted: G. Olms, Hildesheim, 1970), pp 321–324

    Google Scholar 

  • Frisch KR (1955) The logarithmic potentical method of convex programming. Memorandum, University Institute of Economics, Oslo

    Google Scholar 

  • Fulkerson D, Wolfe P (1962) An algorithm for scaling matrices. SIAM Rev 4:142–146

    MathSciNet  Google Scholar 

  • Gale D, Kuhn HW, Tucker AW (1951) Linear programming and the theory of games. In: Koopmans T (ed) Activity analysis of production and allocation. Wiley, New York, pp 317–329

    Google Scholar 

  • Gass SI (1985) Linear programming: methods and applications. McGraw-Hill, New York

    MATH  Google Scholar 

  • Gass SI, Saaty T (1955) The computational algorithm for the parametric objective function. Nav Res Logist Q 2:39–45

    MathSciNet  Google Scholar 

  • Gay DM (1978) On combining the schemes of Reid and Saunders for sparse LP bases. In: Duff IS, Stewart GW (eds) Sparse matrix proceedings. SIAM, Philadelphia, pp 313–334

    Google Scholar 

  • Gay D (1985) Electronic mail distribution of linear programming test problems. COAL Newsl 13:10–12

    Google Scholar 

  • Gay DM (1987) A variant of Karmarkar’s linear programming algorithm for problems in standard form. Math Program 37:81–90

    MATH  MathSciNet  Google Scholar 

  • Geoffrion AM (1972) Generalized Benders decomposition. JOTA 10:137–154

    MathSciNet  Google Scholar 

  • George A, Liu W-H (1981) Computing solution of large sparse positive definite systems. Prentice-Hall, Engleewood Cliffs

    Google Scholar 

  • Gill PE, Murray W (1973) A numerically stable form of the simplex algorithm. Linear Algebra Appl 7:99–138

    MATH  MathSciNet  Google Scholar 

  • Gill PE, Murray W, Saunders MA, Tomlin JA, Wright MH (1985) On projected Newton barrier methods for linear programming and an equivalence to Karmarkar’s projected method. Technical report SOL 85–11, Department of Operations Research, Stanford University

    Google Scholar 

  • Gill PE, Murray W, Saunders MA, Tomlin JA, Wright MH (1986) On projected Newton methods for linear programming and an equivalence to Karmarkar’s projected method. Math Program 36:183–209

    MATH  MathSciNet  Google Scholar 

  • Gill PE, Murray W, Saunders MA, Wright MH (1987) Maintaining LU factors of a general sparse matrix. Linear Algebra Appl 88/89:239–270

    MathSciNet  Google Scholar 

  • Gill PE, Murray W, Saunders MA, Wright MH (1989) A practical anti-cycling procedure for linearly constrainted optimization. Math Program 45:437–474

    MATH  MathSciNet  Google Scholar 

  • Goldfarb D (1977) On the Bartels-Golub decomposition for linear programming bases. Math Program 13:272–279

    MATH  MathSciNet  Google Scholar 

  • Goldfarb D, Reid JK (1977) A practicable steepest-edge simplex algorithm. Math Program 12:361–371

    MATH  MathSciNet  Google Scholar 

  • Goldman AJ, Tucker AW (1956a) Polyhedral convex cones. In: Kuhn HW, Tucker AW (eds) Linear inequalities and related systems. Annals of mathmatical studies, vol 38. Princeton University Press, Princeton, pp 19–39

    Google Scholar 

  • Goldman AJ, Tucker AW (1956b) Theory of linear programming. In: Kuhn HW, Tucker AW (eds) Linear inequalities and related systems. Annals of mathmatical studies, vol 38. Princeton University Press, Princeton, pp 53–97

    Google Scholar 

  • Golub GH (1965) Numerical methods for solving linear least squares problems. Numer Math 7:206–216

    MATH  MathSciNet  Google Scholar 

  • Golub GH, Van Loan CF (1989) Matrix computations, 2edn. The Johns Hopkins University Press, Baltimore

    Google Scholar 

  • Gomory AW (1958) Outline of an algorithm for integer solutions to linear programs. Bull Am Math Soc 64:275–278

    MATH  MathSciNet  Google Scholar 

  • Gonzaga CC (1987) An algorithm for solving linear programming problems in O(n 3 L) operations. Technical Report UCB/ERL M87/10, Electronics Research Laboratory, University of California, Berkeley

    Google Scholar 

  • Gonzaga CC (1990) Convergence of the large step primal affine-scaling algorithm for primal non-degenerate linear problems. Technical report, Department of Systems Engineering and Computer Sciences, COPPE-Federal University of Riode Janeiro

    Google Scholar 

  • Gould N, Reid J (1989) New crash procedure for large systems of linear constraints. Math Program 45:475–501

    MATH  MathSciNet  Google Scholar 

  • Greenberg HJ (1978) Pivot selection tactics. In: Greenberg HJ (ed) Design and implementation of optimization software. Sijthoff and Noordhoff, Alphen aan den Rijn, pp 109–143

    Google Scholar 

  • Greenberg HJ, Kalan J (1975) An exact update for Harris’ tread. Math Program Study 4:26–29

    MathSciNet  Google Scholar 

  • Guerrero-Garcia P, Santos-Palomo A (2005) Phase I cycling under the most-obtuse-angle pivot rule. Eur J Oper Res 167:20–27

    MATH  MathSciNet  Google Scholar 

  • Guerrero-Garcia P, Santos-Palomo A (2009) A deficient-basis dual counterpart of Pararrizos, Samaras ans Stephanides’ primal-dual simplex-type algorithm. Optim Methods Softw 24:187–204

    MATH  MathSciNet  Google Scholar 

  • Güler O, Ye Y (1993) Convergence behavior of interior-point algorithms. Math Program 60:215–228

    MATH  Google Scholar 

  • Hadley G (1972) Linear programming. Addison-Wesley, Reading

    Google Scholar 

  • Hager WW (2002) The dual active set algorithm and its application to linear programming. Comput Optim Appl 21:263–275

    MATH  MathSciNet  Google Scholar 

  • Hall LA, Vanderbei RJ (1993) Two-third is sharp for affine scaling. Oper Res Lett 13:197–201

    MATH  MathSciNet  Google Scholar 

  • Hamming RW (1971) Introduction to applied numerical analysis. McGraw-Hill, New York

    MATH  Google Scholar 

  • Harris PMJ (1973) Pivot selection methods of the Devex LP code. Math Program 5:1–28

    MATH  Google Scholar 

  • Hattersley B, Wilson J (1988) A dual approach to primal degeneracy. Math Program 42:135–145

    MATH  MathSciNet  Google Scholar 

  • He X-C, Sun W-Y (1991) An intorduction to generalized inverse matrix (in Chinese). Jiangsu Science and Technology Press, Nanjing

    Google Scholar 

  • Hellerman E, Rarick DC (1971) Reinversion with the preassigned pivot procedure. Math Program 1:195–216

    MATH  MathSciNet  Google Scholar 

  • Hellerman E, Rarick DC (1972) The partitioned preassigned pivot procedure. In: Rose DJ, Willouhby RA (eds) Sparse matrices and their applications. Plenum, New York, pp 68–76

    Google Scholar 

  • Hertog DD, Roos C (1991) A survey of search directions in interior point methods for linear programming. Math Program 52:481–509

    MATH  Google Scholar 

  • Hoffman AJ (1953) Cycling in the simplex algorithm. Technical report 2974, National Bureau of Standards

    Google Scholar 

  • Hu J-F (2007) A note on “an improved initial basis for the simplex algorithm”. Comput Oper Res 34:3397–3401

    MATH  MathSciNet  Google Scholar 

  • Hu J-F, Pan P-Q (2006) A second note on ‘A method to solve the feasible basis of LP’ (in Chinese). Oper Res Manag Sci 15:13–15

    MathSciNet  Google Scholar 

  • Hu J-F, Pan P-Q (2008a) Fresh views on some recent developments in the simplex algorithm. J Southeast Univ 24:124–126

    MATH  MathSciNet  Google Scholar 

  • Hu J-F, Pan P-Q (2008b) An efficient approach to updating simplex multipliers in the simplex algorithm. Math Program Ser A 114:235–248

    MATH  MathSciNet  Google Scholar 

  • Jansen B, Terlakey T, Roos C (1994) The theory of linear programming: skew symmetric self-dual problems and the central path. Optimization 29:225–233

    MATH  MathSciNet  Google Scholar 

  • Jansen B, Roos C, Terlaky T (1996) Target-following methods for linear programming. In: Terlaky T (ed) Interior point methods of mathematical programming. Kluwer, Dordrecht

    Google Scholar 

  • Jeroslow R (1973) The simplex algorithm with the pivot rule of maximizing criterion improvement. Discret Appl Math 4:367–377

    MATH  MathSciNet  Google Scholar 

  • Kalantari B (1990) Karmarkar’s algorithm with improved steps. Math Program 46:73–78

    MATH  MathSciNet  Google Scholar 

  • Kallio M, Porteus EL (1978) A class of methods for linear programming. Math Program 14:161–169

    MATH  MathSciNet  Google Scholar 

  • Kantorovich LV (1960) Mathematical methods in the organization and planning of production. Manag Sci 6:550–559. Original Russian version appeared in 1939

    Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  • Karmarkar N, Ramakrishnan K (1985) Further developments in the new polynomial-time algorithm for linear progrmming. In: Talk given at ORSA/TIMES national meeting, Boston, Apr 1985

    Google Scholar 

  • Khachiyan L (1979) A polynomial algorithm in linear programming. Doklady Academiia Nauk SSSR 244:1093–1096

    MATH  MathSciNet  Google Scholar 

  • Kirillova FM, Gabasov R, Kostyukova OI (1979) A method of solving general linear programming problems. Doklady AN BSSR (in Russian) 23:197–200

    MATH  Google Scholar 

  • Klee V (1965) A class of linear problemminng problems requiring a larger number of iterations. Numer Math 7:313–321

    MATH  MathSciNet  Google Scholar 

  • Klee V, Minty GJ (1972) How good is the simplex algorithm? In: Shisha O (ed) Inequalities-III. Academic, New York, pp 159–175

    Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  • Koberstein A, Suhl UH (2007) Progress in the dual simplex method for large scale LP problems: practical dual phase 1 algorithms. Comput Optim Appl 37:49–65

    MATH  MathSciNet  Google Scholar 

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

    Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  • Kortanek KO, Shi M (1987) Convergence results and numerical experiments on a linear programming hybrid algorithm. Eur J Oper Res 32:47–61

    MATH  MathSciNet  Google Scholar 

  • Kostina E (2002) The long step rule in the bounded-variable dual simplex method: numerical experiments. Math Methods Oper Res 55:413–429

    MATH  MathSciNet  Google Scholar 

  • Kotiah TCT, Steinberg DI (1978) On the possibility of cycling with the simplex method. Oper Res 26:374–376

    Google Scholar 

  • Kuhn HW, Quandt RE (1953) An experimental study of the simplex method. In: Metropolis NC et al (eds) Eperimental arithmetic, high-speed computing and mathematics. Proceedings of symposia in applied mathematics XV. American Mathematical Society, Providence, pp 107–124

    Google Scholar 

  • Land AH, Doig AG (1960) An automatic method of solving discrete programming problems. Econometrica 28:497–520

    MATH  MathSciNet  Google Scholar 

  • Leichner SA, Dantzig GB, Davis JW (1993) A strictly, improving linear programming phase I algorithm. Ann Oper Res 47:409–430

    MathSciNet  Google Scholar 

  • Lemke CE (1954) The dual method of solving the linear programming problem. Nav Res Logist Q 1:36–47

    MathSciNet  Google Scholar 

  • Li W (2004) A note on two direct methods in linear programming. Eur J Oper Res 158:262–265

    MATH  Google Scholar 

  • Li C, Pan P-Q, Li W (2002) A revised simplex algorithm based on partial pricing pivotrule (in Chinese). J Wenzhou Univ 15:53–55

    Google Scholar 

  • Li W, Guerrero-Garcia P, Santos-Palomo A (2006a) A basis-deficiency-allowing primal phase-1 algorithm using the most-obtuse-angle column rule. Comput Math Appl 51:903–914

    MATH  MathSciNet  Google Scholar 

  • Li W, Pan P-Q, Chen G (2006b) A combined projected gradient algorithm for linear programming. Optim Methods Softw 21:541–550

    MATH  MathSciNet  Google Scholar 

  • Llewellyn RW (1964) Linear programming. Holt, Rinehart and Winston, New York

    Google Scholar 

  • Luo Z-Q, Wu S (1994) A modified predictor-corrector method for linear programming. Comput Optim Appl 3:83–91

    MATH  MathSciNet  Google Scholar 

  • Lustig IJ (1990) Feasibility issues in a prinal-dual interior-point method for linear programming. Math Program 49:145–162

    MATH  MathSciNet  Google Scholar 

  • Lustig IJ, Marsten RE, Shanno DF (1991) Computational exeperience with a primal-dual interior point method for linear programming. Linear Algebra Appl 152:191–222

    MATH  MathSciNet  Google Scholar 

  • Lustig IJ, Marsten RE, Shanno DF (1992) On implementing Mehrotras’s predictor-corrector interior-point for linear programming. SIAM J Optim 2:435–449

    MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  • Markowitz HM (1957) The elimination form of the inverse and its application to linear programming. Manag Sci 3:255–269

    MATH  MathSciNet  Google Scholar 

  • Maros I (1986) A general phase-1 method in linear programming. Eur J Oper Res 23:64–77

    MATH  MathSciNet  Google Scholar 

  • Maros I (2003a) A generalied dual phase-2 simplex algorithm. Eur J Oper Res 149:1–16

    MATH  MathSciNet  Google Scholar 

  • Maros I (2003b) Computational techniques of the simplex method. International series in operations research and management, vol 61. Kluwer, Boston

    Google Scholar 

  • Maros I, Khaliq M (2002) Advances in design and implementation of optimization software. Eur J Oper Res 140:322–337

    MATH  MathSciNet  Google Scholar 

  • Marshall KT, Suurballe JW (1969) A note on cycling in the simplex method. Nav Res Logist Q 16:121–137

    MATH  Google Scholar 

  • Martin RK (1999) Large scale linear and integer optimization: a unified approach. Kluwer, Boston

    MATH  Google Scholar 

  • Mascarenhas WF (1997) The affine scaling algorithm fails for λ = 0. 999. SIAM J Optim 7:34–46

    MATH  MathSciNet  Google Scholar 

  • McShane KA, Monma CL, Shanno DF (1989) An implementation of a primal-dual method for linear programming. ORSA J Comput 1:70–83

    MATH  Google Scholar 

  • Megiddo N (1986a) Introduction: new approaches to linear programming. Algorithmca 1:387–394. (Special issue)

    MATH  MathSciNet  Google Scholar 

  • Megiddo N (1986b) A note on degenerach in linear programming. Math Program 35:365–367

    MATH  MathSciNet  Google Scholar 

  • Megiddo N (1989) Pathways to the optimal set in linear programming. In: Megiddo N (ed) Progress in mathematical programming. Springer, New York, pp 131–158

    Google Scholar 

  • Megiddo N, Shub M (1989) Boundary behavior of interior point algorithm in linear programming. Math Oper Res 14:97–146

    MATH  MathSciNet  Google Scholar 

  • Mehrotra S (1991) On finding a vertex solution using interior point methods. Linear Algebra Appl 152:233–253

    MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  • Mizuno S, Todd MJ, Ye Y (1993) On adaptive-step primal-dual interior-point algorithms for linear programming. Math Oper Res 18:964–981

    MATH  MathSciNet  Google Scholar 

  • Monteiro RDC, Adler I (1989) Interior path following primal-dual algorithms: Part I: linear programming. Math Program 44:27–41

    MATH  MathSciNet  Google Scholar 

  • Murtagh BA (1981) Advances in linear programming: computation and practice. McGraw-Hill, New York/London

    Google Scholar 

  • Murtagh BA, Saunders MA (1978) Large-scale linearly constrained optimization. Math Program 14:41–72

    MATH  MathSciNet  Google Scholar 

  • Murtagh BA, Saunders MA (1998) MINOS 5.5 user’s guid. Technical report SOL 83-20R, Department of Engineering Economics Systems & Operations Research, Stanford University, Stanford

    Google Scholar 

  • Murty KG (1983) Linear programming. Wiley, New York

    MATH  Google Scholar 

  • Nazareth JL (1987) Computer solutions of linear programs. Oxford University Press, Oxford

    Google Scholar 

  • Nazareth JL (1996) The implementation of linear programming algorithms based on homotopies. Algorithmica 15:332–350

    MATH  MathSciNet  Google Scholar 

  • Nemhauser GL (1994) The age of optimizaiton: solving large-scale real-world problems. Oper Res 42:5–13

    MATH  Google Scholar 

  • Nemhauser GL, Wolsey LA (1999) Integer and combinatorial optimization. Wiley, New York

    MATH  Google Scholar 

  • Nocedal J, Wright SJ (1999) Numerical optimization. Springer, Berlin

    MATH  Google Scholar 

  • Ogryczak W (1988) The simplex method is not always well behaved. Linear Algebra Appl 109:41–57

    MATH  MathSciNet  Google Scholar 

  • Orchard-Hays W (1954) Background development and extensions of the revised simplex method. Report RM 1433, The Rand Corporation, Santa Monica

    Google Scholar 

  • Orchard-Hays W (1956) Evolution of computer codes for linear programming. Paper P-810, The RAND Corporation, p 2224

    Google Scholar 

  • Orchard-Hays W (1971) Advanced linear programming computing techniques. McGraw-Hill, New York

    Google Scholar 

  • Padberg MW (1995) Linear optimization and extensions. Springer, Berlin

    MATH  Google Scholar 

  • Pan P-Q (1982) Differential equaltion methods for unconstarained optimization (in Chinese). Numer Math J Chin Univ 4:338–349

    MATH  Google Scholar 

  • Pan P-Q (1990) Practical finite pivoting rules for the simplex method. OR Spektrum 12:219–225

    Google Scholar 

  • Pan P-Q (1991) Simplex-like method with bisection for linear programming. Optimization 22:717–743

    MATH  MathSciNet  Google Scholar 

  • Pan P-Q (1992a) New ODE methods for equality constrained optimization (I) – equations. J Comput Math 10:77–92

    MATH  MathSciNet  Google Scholar 

  • Pan P-Q (1992b) New ODE methods for equality constrained optimization (II) – algorithms. J Comput Math 10:129–146

    MATH  MathSciNet  Google Scholar 

  • Pan P-Q (1992c) Modification of Bland’s pivoting rule (in Chinese). Numer Math 14:379–381

    MATH  Google Scholar 

  • Pan P-Q (1994a) A variant of the dual pivot rule in linear programming. J Inf Optim Sci 15:405–413

    MATH  Google Scholar 

  • Pan P-Q (1994b) Composite phase-1 methods without measuring infeasibility. In: Yue M-Y (ed) Theory of optimization and its applications. Xidian University Press, Xian, pp 359–364.

    Google Scholar 

  • Pan P-Q (1994c) Ratio-test-free pivoting rules for the bisection simplex method. In: Proceedings of national conference on decision making science, Shangrao, pp 24–29

    Google Scholar 

  • Pan P-Q (1994d) Ratio-test-free pivoting rules for a dual phase-1 method. In: Xiao S-T, Wu F (eds) Proceeding of the third conference of Chinese SIAM. Tsinghua University press, Beijing, pp 245–249

    Google Scholar 

  • Pan P-Q (1995) New non-monotone procedures for achieving dual feasibility. J Nanjing Univ Math Biquarterly 12:155–162

    MATH  Google Scholar 

  • Pan P-Q (1996a) A modified bisection simplex method for linear programming. J Comput Math 14:249–255

    MATH  MathSciNet  Google Scholar 

  • Pan P-Q (1996b) New pivot rules for achieving dual feasibility. In: Wei Z (ed) Theory and applications of OR. Proceedings of the fifth conference of Chinese OR society, Xian, 10–14 Oct 1996. Xidian University Press, Xian, pp 109–113

    Google Scholar 

  • Pan P-Q (1996c) Solving linear programming problems via appending an elastic constraint. J Southeast Univ (English edn) 12:253–265

    Google Scholar 

  • Pan P-Q (1997) The most-obtuse-angle row pivot rule for achieving dual feasibility in linear programming: a computational study. Eur J Oper Res 101:164–176

    MATH  Google Scholar 

  • Pan P-Q (1998a) A dual projective simplex method for linear programming. Comput Math Appl 35:119–135

    MATH  MathSciNet  Google Scholar 

  • Pan P-Q (1998b) A basis-deficiency-allowing variation of the simplex method. Comput Math Appl 36:33–53

    MATH  Google Scholar 

  • Pan P-Q (1999a) A new perturbation simplex algorithm for linear programming. J Comput Math 17:233–242

    MATH  MathSciNet  Google Scholar 

  • Pan P-Q (1999b) A projective simplex method for linear programming. Linear Algebra Appl 292:99–125

    MATH  MathSciNet  Google Scholar 

  • Pan P-Q (2000a) A projective simplex algorithm using LU decomposition. Comput Math Appl 39:187–208

    MATH  MathSciNet  Google Scholar 

  • Pan P-Q (2000b) Primal perturbation simplex algorithms for linear programming. J Comput Math 18:587–596

    MATH  MathSciNet  Google Scholar 

  • Pan P-Q (2000c) On developments of pivot algorithms for linear programming. In: Proceedings of the sixth national conference of operations research society of China, Changsha, 10–15 Oct 2000. Global-Link Publishing, Hong Kong, pp 120–129

    Google Scholar 

  • Pan P-Q (2004) A dual projective pivot algorithm for linear programming. Comput Optim Appl 29:333–344

    MATH  MathSciNet  Google Scholar 

  • Pan P-Q (2005) A revised dual projective pivot algorithm for linear programming. SIAM J Optim 16:49–68

    MATH  MathSciNet  Google Scholar 

  • Pan P-Q (2008a) A largest-distance pivot rule for the simplex algorithm. Eur J Oper Res 187:393–402

    MATH  Google Scholar 

  • Pan P-Q (2008b) A primal deficient-basis algorithm for linear programming. Appl Math Comput 198:898–912

    Google Scholar 

  • Pan P-Q (2008c) Efficient nested pricing in the simplex algorithm. Oper Res Lett 38:309–313

    Google Scholar 

  • Pan P-Q (2010) A fast simplex algorithm for linear programming. J Comput Math 28(6):837–847

    MATH  MathSciNet  Google Scholar 

  • Pan P-Q (2013) An affine-scaling pivot algorithm for linear programming. Optimization 62: 431–445

    MATH  MathSciNet  Google Scholar 

  • Pan P-Q, Li W (2003) A non-monotone phase-1 method in linear programming. J Southeast Univ (English edn) 19:293–296

    MathSciNet  Google Scholar 

  • Pan P-Q, Ouiang Z-X (1993) Two variants of the simplex algorithm (in Chinese). J Math Res Expo 13(2):274–275

    Google Scholar 

  • Pan P-Q, Ouyang Z-X (1994) Moore-Penrose inverse simplex algorithms based on successive linear subprogramming approach. Numer Math 3:180–190

    MATH  MathSciNet  Google Scholar 

  • Pan P-Q, Pan Y-P (2001) A phase-1 approach to the generalized simplex algorithm. Comput Math Appl 42:1455–1464

    MATH  MathSciNet  Google Scholar 

  • Pan P-Q, Li W, Wang Y (2004) A phase-1 algorithm using the most-obtuse-angle rule for the basis-deficiency-allowing dual simplex method. OR Trans 8:88–96

    Google Scholar 

  • Pan P-Q, Li W, Cao J (2006a) Partial pricing rule simplex method with deficient basis. Numer Math J Chin Univ (English series) 15:23–30

    MATH  MathSciNet  Google Scholar 

  • Pan P-Q, Hu J-F, Li C (2006b) Feasible region contraction interior point algorithm. Appl Math Comput 182:1361–1368

    MATH  MathSciNet  Google Scholar 

  • Papadimitriou CH, Steiglitz K (1982) Combinatorial optimization: algorithms and complexity. Prentice-Hall, New Jersey

    MATH  Google Scholar 

  • Perold AF (1980) A degeneracy exploiting LU factorization for the simplex method. Math Program 19:239–254

    MATH  MathSciNet  Google Scholar 

  • Powell MJD (1989) A tolerant algorithm for linearly constrained optimization calculations. Math Program 45:547–566

    MATH  Google Scholar 

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

    MATH  Google Scholar 

  • Rockafellar RT (1997) Convext analysis. Princeton University Press, Princeton

    Google Scholar 

  • Roos C (1990) An exponential example for Terlaky’s pivoting rule for the criss-cross simplex method. Math Program 46:79–84

    MATH  MathSciNet  Google Scholar 

  • Roos C, Vial J-Ph (1992) A polynomial method of approximate centers for linear programming. Math Program 54:295–305

    MATH  MathSciNet  Google Scholar 

  • Roos C, Terlaky T, Vial J-P (1997) Theory and algorithms for linear programming. Wiley, Chichester

    Google Scholar 

  • Rothenberg RI (1979) Linear programming. North-Holland, New York

    MATH  Google Scholar 

  • Ryan D, Osborne M (1988) On the solution of highly degenerate linear problems. Math Program 41:385–392

    MATH  MathSciNet  Google Scholar 

  • Saigal R (1995) Linear programming. Kluwer, Boston

    MATH  Google Scholar 

  • Santos-Palomo A (2004) The sagitta method for solving linear problems. Eur J Oper Res 157:527–539

    MATH  MathSciNet  Google Scholar 

  • Saunders MA (1972) Large scale linear programming using the Cholesky factorization. Technical report STAN-CS-72-152, Stanford University

    Google Scholar 

  • Saunders MA (1973) The complexity of LU updating in the simplex method. In: Andersen R, Brent R (eds) The complexity of computational problem solving. University Press, St. Lucia, pp 214–230

    Google Scholar 

  • Saunders MA (1976) A fast and stable implementation of the simplex method using Bartels-Golub updating. In: Bunch J, Rose D (eds) Sparse matrix computation. Academic, New York, pp 213–226

    Google Scholar 

  • Schrijver A (1986) Theory of linear and integer programming. Wiley, Chichester

    MATH  Google Scholar 

  • Shanno DF, Bagchi A (1990) A unified view of interior point methods for linear programming. Ann Oper Res 22:55–70

    MATH  MathSciNet  Google Scholar 

  • Shen Y, Pan P-Q (2006) Dual besiction simplex algorithm (in Chinese). In: Preceedings of the national conference of operatios research society of China, Shenzhen. Globa-Link Informatics, Hong Kong, pp 168–174

    Google Scholar 

  • Shi Y, Pan P-Q (2011) Higher order iteration schemes for unconstrained optimization. Am J Oper Res 1:73–83

    Google Scholar 

  • Smale S (1983a) On the average number of steps of the simplex method of linear programming. Math Program 27:241–262

    MATH  MathSciNet  Google Scholar 

  • Smale S (1983b) The problem of the average speed of the simplex method. In: Bachem A, Grotschel M, Korte B, (eds) Mathematical programming, the state of the art. Springer, Berlin, pp 530–539

    Google Scholar 

  • Srinath LS (1982) Linear programming: principles and applications. Affiliated East-West Press, New Delhi

    Google Scholar 

  • Suhl UH (1994) Mathematical optimization system. Eur J Oper Res 72:312–322

    MATH  Google Scholar 

  • Suhl LM, Suhl UH (1990) Computing sparse LU factorization for large-scale linear programming bases. ORSA J Comput 2:325–335

    MATH  Google Scholar 

  • Suhl LM, Suhl UH (1993) A fast LU-update for linear programming. Ann Oper Res 43:33–47

    MATH  MathSciNet  Google Scholar 

  • Sun W, Yuan Y-X (2006) Optimization theory an methods: nonlinear programming. Springer, New York

    Google Scholar 

  • Swietanowaki A (1998) A new steepest edge approximation for the simplex method for linear programming. Comput Optim Appl 10:271–281

    MathSciNet  Google Scholar 

  • Taha H (1975) Integer programming: theory, applications, and computations. Academic, Orlando

    MATH  Google Scholar 

  • Talacko J V, Rockefeller RT (1960) A Compact Simplex Algorithm and a Symmetric Algorithm for General Linear Programs. Unpublished paper, Marquette University

    Google Scholar 

  • Tanabe K (1977) A geometric method in non-linear programming. Technical Report 23343-AMD780, Brookhaven National Laboratory, New York

    Google Scholar 

  • Tanabe K (1990) Centered Newton method for linear prgramming: interior and ‘exterior’ point method, (in Japannese). In: Tone K (ed) New methods for linear programming 3. The Institute of Statistical Mathematics, Tokeo, pp 98–100

    Google Scholar 

  • Tapia RA, Zhang Y (1991) An optimal-basis identification technique for interior-point linear programming algorithms. Linear Algebra Appl 152:343–363

    MATH  MathSciNet  Google Scholar 

  • Terlaky T (1985) A covergent criss-cross method. Math Oper Stat Ser Optim 16:683–690

    MATH  MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

  • Terlaky T (ed) (1996) Interior point methods of mathematical programming. Kluwer, Dordrecht

    MATH  Google Scholar 

  • Todd MJ (1982) An implementation of the simplex method for linear programming problems with variable upper bounds. Math Program 23:23–49

    Google Scholar 

  • Todd MJ (1983) Large scale linear programming: geometry, working bases and factorizations. Math Program 26:1–23

    MathSciNet  Google Scholar 

  • Tomlin JA (1972) Modifying trangular factors of the basis in the simplex method. In: Rose DJ, Willoughby RA (eds) Sparse matrices and applications. Plenum, New York

    Google Scholar 

  • Tomlin JA (1974) On pricing and backward transformation in linear programming. Math Program 6:42–47

    MATH  MathSciNet  Google Scholar 

  • Tomlin JA (1975) On scaling linear programming problems. Math Program Study 4:146–166

    MathSciNet  Google Scholar 

  • Tomlin JA (1987) An experimental approach to Karmarkar’s projective method, for linear programming. Math Program 31:175–191

    MATH  Google Scholar 

  • Tsuchiya T (1992) Global convergence property of the affine scaling method for primal degenerate linear programming problems. Math Oper Res 17:527–557

    MATH  MathSciNet  Google Scholar 

  • Tsuchiya T, Muramatsu M (1995) Global convergence of a long-step affine-scaling algorithm for degenrate linear programming problems. SIAM J Optim 5:525–551

    MATH  MathSciNet  Google Scholar 

  • Tucker AW (1956) Dual systems of homegeneous linear relations. In: Kuhn HW, Tucker AW, Dantzig GB (eds) Linear inequalities and related systems. Princeton University Press, Princeton, pp 3–18

    Google Scholar 

  • Turner K (1991) Computing projections for the Karmarkar algorithtm. Linear Algebra Appl 152:141–154

    MATH  MathSciNet  Google Scholar 

  • Vanderbei RJ, Lagarias JC (1990) I.I. Dikin’s convergence result for the affine-scaling algorithm. Contemp Math 114:109–119

    MathSciNet  Google Scholar 

  • Vanderbei RJ, Meketon M, Freedman B (1986) A modification of Karmarkars linear programming algorithm. Algorithmica 1:395–407

    MATH  MathSciNet  Google Scholar 

  • Vemuganti RR (2004) On gradient simplex methods for linear programs. J Appl Math Decis Sci 8:107–129

    MATH  MathSciNet  Google Scholar 

  • Wang Z (1987) A conformal elimination-free algorithm for oriented matroid programming. Chin Ann Math 8(B1):16–25

    Google Scholar 

  • Wilkinson JH (1971) Moden error analysis. SIAM Rev 13:548–568

    MATH  MathSciNet  Google Scholar 

  • Wolfe P (1963) A technique for resolving degeneracy in linear programming. J Oper Res Soc 11:205–211

    MATH  MathSciNet  Google Scholar 

  • Wolfe P (1965) The composite simplex algorithm. SIAM Rev 7:42–54

    MATH  MathSciNet  Google Scholar 

  • Wolsey L (1998) Integer programming. Wiley, New York

    MATH  Google Scholar 

  • Wright SJ (1997) Primal-dual interior-point methods. SIAM, Philadelphia

    MATH  Google Scholar 

  • Xu X, Ye Y (1995) A generalized homogeneous and self-dual algorithm for linear prgramming. Oper Res Lett 17:181–190

    MATH  MathSciNet  Google Scholar 

  • Xu X, Hung P-F, Ye Y (1996) A simplified homogeneous and self-dual linear programming algorithm and its implementation. Ann Oper Res 62:151–171

    MATH  MathSciNet  Google Scholar 

  • Yan W-L, Pan P-Q (2001) Improvement of the subproblem in the bisection simplex algorithm (in Chinese). J Southeast Univ 31:324–241

    MathSciNet  Google Scholar 

  • Yan A, Pan P-Q (2005) Variation of the conventional pivot rule and the application in the deficient basis algorithm (in Chinese). Oper Res Manag Sci 14:28–33

    Google Scholar 

  • Yan H-Y, Pan P-Q (2009) Most-obtuse-angle criss-cross algorithm for linear programming (in Chinese). Numer Math J Chin Univ 31:209–215

    MathSciNet  Google Scholar 

  • Yang X-Y, Pan P-Q (2006) Most-obtuse-angle dual relaxation algorithm (in Chinese). In: Preceedings of the national conference of operatios research society of China, Shenzhen. Globa-Link Informatics, Hong Kong, pp 150–155

    Google Scholar 

  • Ye Y (1987) Eliminating columns in the simplex method for linear programming. Technical report SOL 87–14, Department of Operations Research, Stanford Univesity, Stanford

    Google Scholar 

  • Ye Y (1990) A ‘build-down’ scheme for linear probleming. Math Program 46:61–72

    MATH  Google Scholar 

  • Ye Y (1997) Interior point algorithms: theory and analysis. Wiley, New York

    MATH  Google Scholar 

  • Ye Y, Todd MJ, Mizuno S (1994) An \(o(\sqrt{n}l)\)-iteration homogeneous and selfdual linear programming algorithm. Math Oper Res 19:53–67

    MATH  MathSciNet  Google Scholar 

  • Zhang H, Pan P-Q (2008) An interior-point algorithm for linear programming (in Chinese). In: Preceedings of the national conference of operatios research society of China, Nanjing. Globa-Link Informatics, Hong Kong, pp 183–187

    Google Scholar 

  • Zhang J-Z, Xu S-J (1997) Linear programming (in Chinese). Science Press, Bejing

    Google Scholar 

  • Zhang L-H, Yang W-H, Liao L-Z (2013) On an efficient implementation of the face algorithm for linear programming. J Comp Math 31:335–354

    MATH  MathSciNet  Google Scholar 

  • Zhou Z-J, Pan P-Q, Chen S-F (2009) Most-obtuse-angle relaxation algorithm (in Chinese). Oper Res Manag Sci 18:7–10

    Google Scholar 

  • Zionts S (1969) The criss-cross method for solving linear programming problems. Manag Sci 15:420–445

    Google Scholar 

  • Zlatev Z (1980) On some pivotal strategies in Gaussian elimination by sparse technique. SIAM J Numer Anal 17:12–30

    MathSciNet  Google Scholar 

  • Zörnig P (2006) Systematic construction of examples for cycling in the simplex method. Comput Oper Res 33:2247–2262

    MATH  MathSciNet  Google Scholar 

  • Zoutendijk G (1960) Methods of feasible directions. Elsevier, Amsterdam

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Southeast University, Nanjing, China

    Ping-Qi PAN

Authors
  1. Ping-Qi PAN
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

PAN, PQ. (2014). Criss-Cross Simplex Method. In: Linear Programming Computation. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-40754-3_18

Download citation

Publish with us

Policies and ethics

Search

Navigation