Article Outline
Keywords
Synonyms
Introduction
Ziont's Criss-Cross Method
The Least-Index Criss-Cross Method
Other Interpretations
Recursive Interpretation
Lexicographically Increasing List
Other Finite Criss-Cross Methods
FILO)
Most Often Selected Variable Rule
Average Behavior
Best-Case Analysis of Admissible Pivot Methods
Generalizations
Fractional Linear Optimization
Linear Complementarity Problems
Convex Quadratic Optimization
Oriented Matroids
See also
References
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
Bjorner A, Vergnas M LAS, Sturmfels B, White N, Ziegler G (1993) Oriented matroids. Cambridge Univ Press, Cambridge
Bland RG (1977) A combinatorial abstraction of linear programming. J Combin Th B 23:33–57
Bland RG (1977) New finite pivoting rules for the simplex method. Math Oper Res 2:103–107
Chang YY (1979) Least index resolution of degeneracy in linear complementarity problems. Techn Report Dept Oper Res Stanford Univ 14
Cottle R, Pang JS, Stone RE (1992) The linear complementarity problem. Acad Press, New York
Cottle RW, Pang J-S, Venkateswaran V (1987) Sufficient matrices and the linear complementarity problem. Linear Alg Appl 114/115:235–249
Dantzig GB (1963) Linear programming and extensions. Princeton Univ Press, Princeton
Dantzig GB, Orden A, Wolfe P (1955) Notes on linear programming: Part I – The generalized simplex method for minimizing a linear form under linear inequality restrictions. Pacific J Math 5(2):183–195
Fukuda K (1982) Oriented matroid programming. PhD Thesis Waterloo Univ
Fukuda K, Luethi H-J, Namiki M (1997) The existence of a short sequence of admissible pivots to an optimal basis in LP and LCP. ITOR 4:273–284
Fukuda K, Matsui T (1991) On the finiteness of the criss-cross method. Europ J Oper Res 52:119–124
Fukuda K, Namiki M (1994) On extremal behaviors of Murty's least index method. Math Program 64:365–370
Fukuda K, Terlaky T (1992) Linear complementarity and oriented matroids. J Oper Res Soc Japan 35:45–61
Fukuda K, Terlaky T (1997) Criss-cross methods: A fresh view on pivot algorithms. Math Program (B) In: Lectures on Math Program, vol 79. ISMP97, Lausanne, pp 369–396
Fukuda K, Terlaky T (1999) On the existence of short admissible pivot sequences for feasibility and linear optimization problems. Techn Report Swiss Federal Inst Technol
Hertog D Den, Roos C, Terlaky T (1993) The linear complementarity problem, sufficient matrices and the criss-cross method. Linear Alg Appl 187:1–14
Illés T, Szirmai Á, Terlaky T (1999) A finite criss-cross method for hyperbolic programming. Europ J Oper Res 114:198–214
Jensen D (1985) Coloring and duality: Combinatorial augmentation methods. PhD Thesis School OR and IE, Cornell Univ
Klafszky E, Terlaky T (1989) Some generalizations of the criss-cross method for the linear complementarity problem of oriented matroids. Combinatorica 9:189–198
Klafszky E, Terlaky T (1992) Some generalizations of the criss-cross method for quadratic programming. Math Oper Statist Ser Optim 24:127–139
Klee V, Minty GJ (1972) How good is the simplex algorithm? In: Shisha O (ed) Inequalities-III. Acad Press, New York, pp 1159–175
Lemke CE (1968) On complementary pivot theory. In: Dantzig GB, Veinott AF (eds) Mathematics of the Decision Sci Part I. Lect Appl Math 11. Amer Math Soc. Providence, RI, pp 95–114
Lustig I (1987) The equivalence of Dantzig's self-dual parametric algorithm for linear programs to Lemke's algorithm for linear complementarity problems applied to linear programming. SOL Techn Report Dept Oper Res Stanford Univ 87(4)
Murty KG (1974) A note on a Bard type scheme for solving the complementarity problem. Opsearch 11(2–3):123–130
Roos C (1990) An exponential example for Terlaky's pivoting rule for the criss-cross simplex method. Math Program 46:78–94
Terlaky T (1984) Egy új, véges criss-cross módszer lineáris programozási feladatok megoldására. Alkalmazott Mat Lapok 10:289–296 English title: A new, finite criss-cross method for solving linear programming problems. (In Hungarian)
Terlaky T (1985) A convergent criss-cross method. Math Oper Statist Ser Optim 16(5):683–690
Terlaky T (1987) A finite criss-cross method for oriented matroids. J Combin Th B 42(3):319–327
Terlaky T, Zhang S (1993) Pivot rules for linear programming: A survey on recent theoretical developments. Ann Oper Res 46:203–233
Todd MJ (1984) Complementarity in oriented matroids. SIAM J Alg Discrete Meth 5:467–485
Todd MJ (1985) Linear and quadratic programming in oriented matroids. J Combin Th B 39:105–133
Tucker A (1977) A note on convergence of the Ford–Fulkerson flow algorithm. Math Oper Res 2(2):143–144
Valiaho H (1992) A new proof of the finiteness of the criss-cross method. Math Oper Statist Ser Optim 25:391–400
Wang Zh (1985) A conformal elimination free algorithm for oriented matroid programming. Chinese Ann Math 8(B1)
Wang Zh (1991) A modified version of the Edmonds–Fukuda algorithm for LP in the general form. Asia–Pacific J Oper Res 8(1)
Wang Zh (1992) A general deterministic pivot method for oriented matroid programming. Chinese Ann Math B 13(2)
Zhang S (1991) On anti-cycling pivoting rules for the simplex method. Oper Res Lett 10:189–192
Zhang S (1999) New variants of finite criss-cross pivot algorithms for linear programming. Europ J Oper Res 116:607–614
Zionts S (1969) The criss-cross method for solving linear programming problems. Managem Sci 15(7):426–445
Zionts S (1972) Some empirical test of the criss-cross method. Managem Sci 19:406–410
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2008 Springer-Verlag
About this entry
Cite this entry
Terlaky, T. (2008). Criss-Cross Pivoting Rules . In: Floudas, C., Pardalos, P. (eds) Encyclopedia of Optimization. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-74759-0_103
Download citation
DOI: https://doi.org/10.1007/978-0-387-74759-0_103
Publisher Name: Springer, Boston, MA
Print ISBN: 978-0-387-74758-3
Online ISBN: 978-0-387-74759-0
eBook Packages: Mathematics and StatisticsReference Module Computer Science and Engineering