Mathematical Programming

, Volume 79, Issue 1–3, pp 369–395 | Cite as

Criss-cross methods: A fresh view on pivot algorithms

  • Komei Fukuda
  • Tamás Terlaky


Criss-cross methods are pivot algorithms that solve linear programming problems in one phase starting with any basic solution. The first finite criss-cross method was invented by Chang, Terlaky and Wang independently. Unlike the simplex method that follows a monotonic edge path on the feasible region, the trace of a criss-cross method is neither monotonic (with respect to the objective function) nor feasibility preserving. The main purpose of this paper is to present mathematical ideas and proof techniques behind finite criss-cross pivot methods. A recent result on the existence of a short admissible pivot path to an optimal basis is given, indicating shortest pivot paths from any basis might be indeed short for criss-cross type algorithms. The origins and the history of criss-cross methods are also touched upon.


Linear programming Quadratic programming Linear complementarity problems Oriented matroids Pivot rules Criss-cross method Cycling Recursion 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    I. Adler and N. Megiddo, A simplex algorithm whose average number of steps is bounded between two quadratic functions of the smaller dimension,Journal of the Association of Computing Machinery 32 (1985) 891–895.MathSciNetGoogle Scholar
  2. [2]
    D. Avis and V. Chvátal, Notes on Bland’s rule,Mathematical Programming Study 8 (1978) 24–34.Google Scholar
  3. [3]
    D. Avis and K. Fukuda, A pivoting algorithm for convex hulls and vertex enumeration of arrangements and polyhedra,Discrete and Computational Geometry 8 (1992) 295–313.zbMATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    J. Beasley, ed.,Advances in Linear and Integer Programming (Oxford University Press, Oxford, UK, 1996).zbMATHGoogle Scholar
  5. [5]
    R.E. Bixby, The simplex method: It keeps getting better, Lecture at the 14th International Symposium on Mathematical Programming, Amsterdam, The Netherlands, 1991.Google Scholar
  6. [6]
    A. Björner, M. Las Vergnas, B. Sturmfels, N. White and G. Ziegler,Oriented Matroids (Cambridge University Press, Cambridge, MA, 1993).zbMATHGoogle Scholar
  7. [7]
    R.G. Bland, New finite pivoting rules for the simplex method,Mathematics of Operations Research 2 (1977) 103–107.zbMATHMathSciNetGoogle Scholar
  8. [8]
    R.G. Bland, A combinatorial abstraction of linear programming,Journal of Combinatorial Theory, Series B 23 (1977) 33–57.zbMATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    R.G. Bland and M. Las Vergnas, Orientability of matroids,Journal of Combinatorial Theory, Series B 24 (1978) 94–123.zbMATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    K.H. Borgwardt,The Simplex Method: A Probabilistic Analysis, Algorithms and Combinatorics, Vol. 1 (Springer, Berlin, 1987).Google Scholar
  11. [11]
    K. Cameron and J. Edmonds, Existentially polytime theorems, in:Proceedings of the DIMACS Workshop in Polyhedral Combinatorics, 1993.Google Scholar
  12. [12]
    Y.Y. Chang, Least index resolution of degeneracy in linear complementarity problems, Technical Report 79-14, Department of Operations Research, Stanford University, Stanford, CA, 1979.Google Scholar
  13. [13]
    A. Charnes, Optimality and degeneracy in linear programming,Econometrica 20 (1952) 160–170.zbMATHCrossRefMathSciNetGoogle Scholar
  14. [14]
    H.-D. Chen, P.M. Pardalos and M.A. Saunders, The simplex algorithm with a new primal and dual pivot rule,Operations Research Letters 16 (1994) 121–127.zbMATHCrossRefMathSciNetGoogle Scholar
  15. [15]
    V. Chvátal,Linear Programming (W.H. Freeman and Company, San Francisco, 1983).zbMATHGoogle Scholar
  16. [16]
    J. Clausen, A note on Edmonds-Fukuda pivoting rule for the simplex method,European Journal of Operations Research 29 (1987) 378–383.zbMATHCrossRefMathSciNetGoogle Scholar
  17. [17]
    R.W. Cottle, Symmetric dual quadratic programs,Quarterly of Applied Mathematics 21 (1963) 237–243.zbMATHMathSciNetGoogle Scholar
  18. [18]
    R.W. Cottle and G.B. Dantzig, Complementary pivot theory of mathematical programming,Linear Algebra and Its Applications 1 (1968) 103–125.zbMATHCrossRefMathSciNetGoogle Scholar
  19. [19]
    R.W. Cottle, J.-S. Pang and V. Venkateswaran, Sufficient matrices and the linear complementarity problem.Linear Algebra and Its Applications 114/115 (1987) 235–249.Google Scholar
  20. [20]
    R.W. Cottle, J.-S. Pang and R.E. Stone,The Linear Complementarity Problem (Academic Press, New York, 1992).zbMATHGoogle Scholar
  21. [21]
    G.B. Dantzig, Programming in a linear structure,Comptroller, USAF Washington, DC (February 1948).Google Scholar
  22. [22]
    G.B. Dantzig,Linear Programming and Extensions (Princeton University Press, Princeton, NJ, 1963).zbMATHGoogle Scholar
  23. [23]
    G.B. Dantzig, Linear programming: The story about how it began, in: A.H.G. Rinnoy Kan, L.K. Lenstra and A. Schrijver, eds.,History of Mathematical Programming (CWI, North-Holland, Amsterdam, 1991). 19–31.Google Scholar
  24. [24]
    G.B. Dantzig, A. Orden and P. Wolfe, The generalized simplex method for minimizing a linear form under linear inequality restraints,Pacific Journal of Mathematics 5 (1955) 183–195.zbMATHMathSciNetGoogle Scholar
  25. [25]
    J. Edmonds, Exact pivoting, Presentation, ECCO VII, 1994.Google Scholar
  26. [26]
    J. Edmonds, A Helly method for linear programming, presentation, CO94, Amsterdam, 1994.Google Scholar
  27. [27]
    J. Edmonds and K. Fukuda, NP easy and LP theory, Cours postgrade en Recherche Opérationnelle, École Polytechnique Fédérale de Lausanne, Lausanne, 1994.Google Scholar
  28. [28]
    J. Folkman and J. Lawrence, Oriented matroids,Journal of Combinatorial Theory, Series B 25 (1978) 199–236.zbMATHCrossRefMathSciNetGoogle Scholar
  29. [29]
    K. Fukuda, Oriented matroid programming, Ph.D. Thesis, Waterloo University, Waterloo, Ontario, Canada, 1982.Google Scholar
  30. [30]
    K. Fukuda, H.-J. Lüthi and M. Namiki, The existence of a short sequence of admissible pivots to an optimal basis in LP and LCP,ITOR, to appear.Google Scholar
  31. [31]
    K. Fukuda and H.-J. Lüthi, Combinatorial maximum improvement algorithm for LP and LCP, Presented at Franco-Japanese Days, Brest, France, 1995.Google Scholar
  32. [32]
    K. Fukuda and T. Matsui, On the finiteness of the criss-cross method,European Journal of Operations Research 52 (1991) 119–124.zbMATHCrossRefGoogle Scholar
  33. [33]
    K. Fukuda and M. Namiki, On extremal behaviors of Murty’s least index method,Mathematical Programming 64 (1994) 365–370.CrossRefMathSciNetGoogle Scholar
  34. [34]
    K. Fukuda and M. Namiki, Finding all common basis in two matroids,Discrete Applied Mathematics 56 (1995) 231–243.zbMATHCrossRefMathSciNetGoogle Scholar
  35. [35]
    K. Fukuda, M. Namiki and A. Tamura, EP theorems and linear complementarity problems, Technical Report, Institute for Operations Research, ETH-Zentrum, CH-8092 Zürich, Switzerland, 1996.Google Scholar
  36. [36]
    K. Fukuda and T. Terlaky, Linear complementarity and oriented matroids,Journal of the Operational Research Society of Japan 35 (1992) 45–61.zbMATHMathSciNetGoogle Scholar
  37. [37]
    D. Goldfarb, Worst case complexity of the shadow vertex simplex algorithm, Technical Report, Department of Industrial Engineering and Operations Research, Columbia University, 1983.Google Scholar
  38. [38]
    D. Den Hertog, C. Roos and T. Terlaky, The linear complementarity problem, sufficient matrices and the criss-cross method,Linear Algebra and Its Applications 187 (1993) 1–14.zbMATHCrossRefMathSciNetGoogle Scholar
  39. [39]
    T. Illés. Á. Szirmai and T. Terlaky, A finite criss-Cross method for hyperbolic programming, Report No. 96-103, Faculteit der Technische Wiskunde en Informatica, Technische Universiteit Delft, The Netherlands, 1996; also in:European Journal of Operations Research, to appear.Google Scholar
  40. [40]
    D. Jensen, Coloring and duality: Combinatorial augmentation methods, Ph.D. Thesis, School of OR and IE, Cornell University, Ithaca, NY, 1985.Google Scholar
  41. [41]
    G. Kalai and D. Kleitman, A quasi-polynomial bound for the diameter of graphs of polyhedra,Bull. Amer. Math. Soc. 26 (1992) 315–316.zbMATHMathSciNetGoogle Scholar
  42. [42]
    N. Karmarkar, A new polynomial-time algorithm for linear programming,Combinatorica 4 (1984) 373–395.zbMATHCrossRefMathSciNetGoogle Scholar
  43. [43]
    L.G. Khachian, Polynomial algorithms in linear programming,Zhurnal Vichislitelnoj Matematiki i Matematischeskoi Fiziki 20 (1980) 51–68 (in Russian); English translation in:USSR Computational Mathematics and Mathematical Physics 20 (1980) 53–72.Google Scholar
  44. [44]
    E. Klafszky and T. Terlaky, Some generalizations of the criss-cross method for quadratic programming,Math. Oper. und Stat. Ser. Optimization 24 (1992) 127–139.zbMATHMathSciNetGoogle Scholar
  45. [45]
    E. Klafszky and T. Terlaky, Some generalizations of the criss-cross method for the linear complementarity problem of oriented matroids,Combinatorica 9 (1989) 189–198.zbMATHCrossRefMathSciNetGoogle Scholar
  46. [46]
    V. Klee and G.J. Minty, How good is the simplex algorithm? in: O. Shisha, ed.,Inequalities - III (Academic Press, New York, 1972) 159–175.Google Scholar
  47. [47]
    V. Klee and P. Kleinschmidt, Thed-step conjecture and its relatives,Mathematics of Operations Research 12 (1987) 718–755.zbMATHMathSciNetGoogle Scholar
  48. [48]
    M. Kojima, N. Megiddo, T. Noma and A. Yoshise,A Unified Approach to Interior Point Algorithms for Linear Complementarity Problems, Lecture Notes in Computer Science Vol. 538 (Springer, Berlin, 1991).Google Scholar
  49. [49]
    C.E. Lemke and J.T. Howson Jr, Equilibrium points of bimatrix games,SIAM Journal 12 (1964) 413–423.zbMATHMathSciNetGoogle Scholar
  50. [50]
    C.E. Lemke, On complementary pivot theory, in: G.B. Dantzig and A.F. Veinott, eds.,Mathematics of Decision Sciences, Part 1 (AMS, Providence, RI, 1968) 95–114.Google Scholar
  51. [51]
    T.M. Liebling, On the number of iterations of the simplex method, in: R. Henn, H.P. Künzi and H. Schubert, eds.,Methods of Operations Research XVII (Verlag Anton, 1972) 248–264.Google Scholar
  52. [52]
    I. Lustig, The equivalence of Dantzig’s self-dual parametric algorithm for linear programs to Lemke’s algorithm for linear complementarity problems applied to linear programming, Technical Report SOL 87-4, Department of Operations Research, Stanford University, Stanford, California, 1987.Google Scholar
  53. [53]
    A. Mandel, Topology of oriented matroids, Ph.D. Thesis, Waterloo University, Waterloo, Ontario, 1982.Google Scholar
  54. [54]
    W. Morris Jr and M.J. Todd, Symmetry and positive definiteness in oriented matroids, Technical Report No. 631, Cornell University, School of Operations Research and Industrial Engineering, Ithaca, NY, 1984.Google Scholar
  55. [55]
    K.G. Murty, A note on a Bard type scheme for solving the complementarity problem,Opsearch 11 (2–3) (1974) 123–130.MathSciNetGoogle Scholar
  56. [56]
    K.G. Murty,Linear and Combinatorial Programming (Krieger Publishing Company, Malabar, FL, 1976).zbMATHGoogle Scholar
  57. [57]
    K.G. Murty, Computational complexity of parametric linear programming,Mathematical Programming 19 (1980) 213–219.zbMATHCrossRefMathSciNetGoogle Scholar
  58. [58]
    M. Namiki and T. Matsui, Some modifications of the criss-cross method, Research Report, Department of Information Sciences, Tokyo Institute of Technology, Tokyo, 1990.Google Scholar
  59. [59]
    K. Paparrizos, Pivoting rules directing the simplex method through all feasible vertices of Klee-Minty examples,Opsearch 26 (2) (1989) 77–95.zbMATHMathSciNetGoogle Scholar
  60. [60]
    C. Roos, An exponential example for Terlaky’s pivoting rule for the criss-cross simplex method,Mathematical Programming 46 (1990) 78–94.CrossRefMathSciNetGoogle Scholar
  61. [61]
    C. Roos, T. Terlaky and J.-Ph. Vial,Theory and Algorithms for Linear Optimization: An Interior Point Approach (John Wiley & Sons, New York, 1997).zbMATHGoogle Scholar
  62. [62]
    J.E. Strum,Introduction to Linear Programming (Holden-Day, San Francisco, CA, 1972).zbMATHGoogle Scholar
  63. [63]
    T. Terlaky, Egy új, véges criss-cross módszer programozási feladatok megoldására,Alkalmazott Matematikai Lapok 10 (1984) 289–296 (in Hungarian, English title: A new, finite criss-cross method for solving linear programming problems).zbMATHMathSciNetGoogle Scholar
  64. [64]
    T. Terlaky, A convergent criss-cross method,Math. Oper. und Stat. ser. Optimization 16 (5) (1985) 683–690.zbMATHMathSciNetGoogle Scholar
  65. [65]
    T. Terlaky, A finite criss-cross method for oriented matroids,Journal of Combinatorial Theory, Series B 42 (3) (1987) 319–327.zbMATHCrossRefMathSciNetGoogle Scholar
  66. [66]
    T. Terlaky, ed.,Interior Point Methods of Mathematical Programming (Kluwer Academic Publishers, Dordrecht, The Netherlands, 1996).zbMATHGoogle Scholar
  67. [67]
    T. Terlaky and S. Zhang, Pirvot rules for linear programming: A survey on recent theoretical developments,Annals of Operations Research 46 (1993) 203–233.CrossRefMathSciNetGoogle Scholar
  68. [68]
    M.J. Todd, Complementarity in oriented matroids,SIAM Journal on Algebraic and Discrete Mathematics 5 (1984) 467–485.zbMATHMathSciNetCrossRefGoogle Scholar
  69. [69]
    M.J. Todd, Linear and quadratic programming in oriented matroids,Journal of Combinatorial Theory, Series B 39 (1985) 105–133.zbMATHCrossRefMathSciNetGoogle Scholar
  70. [70]
    M.J. Todd, Polynomial expected behavior of a pivoting algorithm for linear complementarity and linear programming problems,Mathematical Programming 35 (1986) 173–192.zbMATHCrossRefMathSciNetGoogle Scholar
  71. [71]
    A. Tucker, A note on convergence of the Ford-Fulkerson flow algorithm,Mathematics of Operations Research 2 (2) (1977) 143–144.zbMATHMathSciNetGoogle Scholar
  72. [72]
    H. Väliaho, A new proof of the finiteness of the criss-cross method,Math. Oper. und Stat. ser. Optimization 25 (1992) 391–400.zbMATHGoogle Scholar
  73. [73]
    H. Väliaho,P *-matrices are just sufficient,Linear Algebra and Its Applications 239 (1996) 103–108.zbMATHMathSciNetGoogle Scholar
  74. [74]
    Zh. Wang, A conformal elimination free algorithm for oriented matroid programming,Chinese Annals of Mathematics 8 (B1) (1987).Google Scholar
  75. [75]
    Zh. Wang, A modified version of the Edmonds-Fukuda algorithm for LP in the general form,Asia-Pacific Journal of Operations Research 8 (1) (1991).Google Scholar
  76. [76]
    Zh. Wang, A general deterministic pivot method for oriented matroid programming,Chinese Annals of Mathematics B 13 (2) (1992).Google Scholar
  77. [77]
    Zh. Wang and T. Terlaky, A general scheme for solving linear complementarity problems in the setting of oriented matroids, in: H.P. Yap, T.H. Ku, E.K. Lloyd and Zh. Wang, eds.,Combinatorics and Graph Theory, Proceedings of the Spring School and International Conference on Combinatorics: SSIC’92, China (World Scientific, Singapore, 1993) 244–255.Google Scholar
  78. [78]
    S.J. Wright,Primal-Dual Interior Point Methods (SIAM Publications, Philadelphia, PA, 1996).Google Scholar
  79. [79]
    N. Zadeh, What is the worst case behavior of the simplex algorithm? Technical Report No. 27, Department of Operations Research, Stanford University, Stanford, CA, 1980.Google Scholar
  80. [80]
    S. Zhang, On anti-cycling pivoting rules for the simplex method,Operations Research Letters 10 (1991) 189–192.zbMATHCrossRefMathSciNetGoogle Scholar
  81. [81]
    S. Zhang, New variants of finite criss-cross pivot algorithms for linear programming, Technical Report, Econometric Institute, Erasmus University Rotterdam, 9707/A, 1997.Google Scholar
  82. [82]
    S. Zionts, The criss-cross method for solving linear programming problems,Management Science 15 (7) (1969) 426–445.Google Scholar
  83. [83]
    S. Zionts, Some empirical test of the criss-cross method,Management Science 19 (1972) 406–410.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© The Mathematical Programming Society, Inc 1997

Authors and Affiliations

  • Komei Fukuda
    • 1
    • 2
  • Tamás Terlaky
    • 3
  1. 1.Department of MathematicsSwiss Federal Institute of TechnologyLausanneSwitzerland
  2. 2.Institute for Operations Research, Swiss Federal Institute of TechnologyZürichSwitzerland
  3. 3.Faculty of Technical Mathematics and InformaticsDelft University of TechnologyDelftThe Netherlands

Personalised recommendations