# Pivot rules for linear programming: A survey on recent theoretical developments

- 376 Downloads
- 52 Citations

## Abstract

The purpose of this paper is to discuss the various pivot rules of the simplex method and its variants that have been developed in the last two decades, starting from the appearance of the minimal index rule of Bland. We are mainly concerned with finiteness properties of simplex type pivot rules. Well known classical results concerning the simplex method are not considered in this survey, but the connection between the new pivot methods and the classical ones, if there is any, is discussed.

In this paper we discuss three classes of recently developed pivot rules for linear programming. The first and largest class is the class of essentially combinatorial pivot rules including minimal index type rules and recursive rules. These rules only use labeling and signs of the variables. The second class contains those pivot rules which can actually be considered as variants or generalizations or specializations of Lemke's method, and so they are closely related to parametric programming. The last class has the common feature that the rules all have close connections to certain interior point methods. Finally, we mention some open problems for future research.

## Keywords

Linear programming simplex method pivot rules cycling recursion minimal index rule parametric programming## Preview

Unable to display preview. Download preview PDF.

## References

- [1]K.M. Anstreicher and T. Terlaky, A monotonic build-up simplex algorithm, Report 91-82. Faculty of Technical Mathematics and Informatics, Delft University of Technology, The Netherlands (1990), to appear in Oper. Res.Google Scholar
- [2]I. Adler and N. Megiddo, A simplex algorithm whose average number of steps is bounded between two quadratic functions of the smaller dimension, J. ACM 32(1985)891–895.CrossRefGoogle Scholar
- [3]D. Avis and V. Chvátal, Notes on Bland's rule, Math. Progr. Study 8(1978)24–34.Google Scholar
- [4]D. Avis and K. Fukuda, A pivoting algorithm for convex hulls and vertex enumeration of arrangements and polyhedra, Discr. Comp. Geom. 8(1992)295–313.CrossRefGoogle Scholar
- [5]R.E. Bixby, Implementing the simplex method: The initial basis, Report TR90-32, Department of Mathematical Sciences, Rice University, TX, USA (1991).Google Scholar
- [6]R.E. Bixby, The simplex method: It keeps getting better, Lecture at the
*14th Int. Symp. on Mathematical Programming*, Amsterdam, The Netherlands (1991).Google Scholar - [7]R.E. Bixby, J.W. Gregory, I.J. Lustig, R.E. Marsten and D.F. Shanno, Very large-scale linear programming: A case study in combining interior point and simplex methods, Oper. Res. 40(1992)885–897.CrossRefGoogle Scholar
- [8]R.G. Bland, New finite pivoting rules for the simplex method, Math. Oper. Res. 2(1977)103–107.CrossRefGoogle Scholar
- [9]R.G. Bland, A combinatorial abstraction of linear programming. J. Combin. Theory (Ser. B) 23(1977)33–57.CrossRefGoogle Scholar
- [10]K.H. Borgwardt,
*The Simplex Method: A Probabilistic Analysis*, Algorithms and Combinatorics, Vol. 1 (Springer, 1987).Google Scholar - [11]E.A. Boyd, Resolving degeneracy in combinatorial linear programes: Steepest edge, steepest ascent and closest ascent, Report, Department of Mathematical Sciences, Rice University, TX, USA (1992).Google Scholar
- [12]N. Cameron, Stationarity in the simplex method. J. Austral. Math. Soc. 43(1987)137–142.CrossRefGoogle Scholar
- [13]Y.Y. Chang, Least index resolution of degeneracy in linear complementarity problems, Technical Report 79-14, Department of Operations Research, Stanford University, Stanford, CA, USA (1979).Google Scholar
- [14]Y.Y. Chang and R.W. Cottle, Least index resolution of degeneracy in quadratic programming, Math. Progr. 18(1980)127–137.CrossRefGoogle Scholar
- [15]S.Y. Chang and K.G. Murty, The steepest descent gravitational method for linear programming, Discr. Appl. Math. 25(1989)211–239.CrossRefGoogle Scholar
- [16]S.S. Chiu and Y. Ye, Simplex method and Karmarkar's algorithm: A unifying structure, Technical Report, Engineering Economic Systems Department, Stanford University, Stanford, CA, USA (1985).Google Scholar
- [17]M. Cirina, Remarks on a recent simplex pivoting rule, Math. Oper. Res. 49(1985)187–199.Google Scholar
- [18]J. Clausen, A note on Edmonds-Fukuda's pivoting rule for the simplex method, Eur. J. Oper. Res. 29(1987)378–383.CrossRefGoogle Scholar
- [19]R.W. Cottle, The principal pivoting method revisited, Math. Progr. Ser. B, 48(1990)369–385.CrossRefGoogle Scholar
- [20]G.B. Dantzig,
*Linear Programming and Extensions*(Princeton University Press, Princeton, NJ, 1963).Google Scholar - [21]H.A. Eiselt and C.-L. Sandblom, External pivoting in the simplex algorithm, Statistica Neerlandica 39(1985)327–341.CrossRefGoogle Scholar
- [22]J. Folkman and J. Lawrence, Oriented matroids, J. Combin. Theory (Ser. B) 25(1978)199–236.CrossRefGoogle Scholar
- [23]J.J.H. Forrest and J.A. Tomlin, Implementing the simplex method for the optimization subroutine library, IBM Sys. J. 31(1992)11–25.CrossRefGoogle Scholar
- [24]K. Fukuda, Oriented matroid programming, Ph.D. Thesis, Waterloo University, Waterloo, Ontario, Canada (1982).Google Scholar
- [25]K. Fukuda and T. Matsui, On the finiteness of the criss-cross method, Eur. J. Oper. Res. 52(1991)119–124.CrossRefGoogle Scholar
- [26]K. Fukuda and M. Namiki, Two extremal behaviors of the criss-cross method for linear complementarity problems, Research Report B-241, Department of Information Sciences, Tokyo Institute of Technology, Tokyo, Japan (1991).Google Scholar
- [27]K. Fukuda and T. Terlaky, A general algorithmic framework for quadratic programming and a generalization of Edmonds-Fukuda's rule as a finite version of the Van de Panne-Whinston method, Preprint, Eötvös University, Budapest (1989).Google Scholar
- [28]K. Fukuda and T. Terlaky, Linear complementarity and oriented matroids, J. Oper. Res. Soc. Japan 35(1992)45–61.Google Scholar
- [29]T. Gal, Degeneracy graphs — Theory and application: A state-of-the-art survey, Report No. 142, Fern Universität Hagen, Germany (1989).Google Scholar
- [30]T. Gal and F. Geue, The use of the TNP-rule to solve various degeneracy problems, Report No. 149a, Fern Universität Hagen, Germany (1990).Google Scholar
- [31]S. Gass and Th. Saaty, The computational algorithm for the parametric objective function, Naval Res. Log. Quarterly 2(1955)39–45.CrossRefGoogle Scholar
- [32]F. Geue, Eine neue Pivotauswahlregel und die durch sie induzierten Teilgraphen des positiven Entartungsgraphen, Report No. 141, Fern Universität Hagen, Germany (1989).Google Scholar
- [33]P.E. Gill, W. Murray, M.A. Saunders and M.H. Wright, A practical anti-cycling procedure for linearly constrained optimization, Math. Progr. 45(1989)437–474.CrossRefGoogle Scholar
- [34]D. Goldfarb, Worst case complexity of the shadow vertex simplex algorithm, Report, Department of Industrial Engineering and Operations Research, Columbia University, New York, USA (1983).Google Scholar
- [35]D. Goldfarb and J.J.H. Forrest, Steepest edge simplex algorithm for linear programming, Math. Progr. 57(1992)341–374.CrossRefGoogle Scholar
- [36]M. Haimovitz, The simplex method is very good! — On the expected number of pivot steps and related properties of random linear programs, Report, Columbia University, New York, USA (1983).Google Scholar
- [37]P.M.J. Harris, Pivot selection methods for the Devex LP code, Math. Progr. 5(1973)1–28.CrossRefGoogle Scholar
- [38]B. Hattersly and J. Wilson, A dual approach to primal degeneracy, Math. Progr. 42(1988)135–176.CrossRefGoogle Scholar
- [39]D. den Hertog, C. Roos and T. Terlaky, The linear complementarity problem, sufficient matrices and the criss-cross method, Lin. Alg. Appl. 187(1993)1–14.CrossRefGoogle Scholar
- [40]D. den Hertog, C. Roos and T. Terlaky, The inverse barrier method for linear programming, Report No. 91-27, Faculty of Technical Mathematics and Informatics, Delft University of Technology, The Netherlands (1991).Google Scholar
- [41]D. Jensen, Coloring and duality: Combinatorial augmentation methods, Ph.D. Thesis, School of OR and IE, Cornell University, Ithaca, NY, USA (1985).Google Scholar
- [42]J.E. Kalan, Machine inspired enhancements of the simplex algorithm, Technical Report CS75001-R, Computer Science Department, Virginia Polytechnical University, Blacksburg, VA, USA (1976).Google Scholar
- [43]N. Karmarkar, A new polynomial-time algorithm for linear programming, Combinatorica 4(1984)373–395.CrossRefGoogle Scholar
- [44]L.G. Khachian, Polynomial algorithms in linear programming, Zhurnal Vichislitelnoj Matematiki i Matematischeskoi Fiziki 20(1980)51–68 [in Russian transl: USSR Comp. Math. Math. Phys. 20(1980)53–72.Google Scholar
- [45]E. Klafszky and T. Terlaky, Remarks on the feasibility problems of oriented matroids, Ann. Univ. Sci. Budapestiensis de Rolando Eötvös Nominatae, Sectio Computatorica, 7(1987)155–157.Google Scholar
- [46]E. Klafszky and T. Terlaky, Variants of the Hungarian method for solving linear programming problems. Math. Oper. und Stat., ser. Optimization 20(1989)79–91.Google Scholar
- [47]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.Google Scholar
- [48]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.CrossRefGoogle Scholar
- [49]E. Klee and G.J. Minty, How good is the simplex algorithm? in:
*Inequalities — III*, ed. O. Shisha (Academic Press, New York, 1972) pp. 159–175.Google Scholar - [50]V. Klee and P. Kleinshcmidt, The
*d*-step conjecture and its relatives, Math. Oper. Res. 12(1987)718–755.CrossRefGoogle Scholar - [51]E. Klotz, Dynamic pricing criteria in linear programming, Technical Report SOL. No. 88-15, Systems Optimization Laboratory, Stanford University, Stanford, CA, USA (1988).Google Scholar
- [52]K.O. Kortanek and J. Zhu, New purification algorithms for linear programming, Naval Res. Log. Quarterly 35(1988)571–583.CrossRefGoogle Scholar
- [53]H.P. Künzi and H. Tzschach, The duoplex-algorithm, Numer. Mathematik 7(1965)222–225.CrossRefGoogle Scholar
- [54]C.E. Lemke, Bimatrix equilibrium points and mathematical programming, Manag. Sci. 11(1965)681–689.CrossRefGoogle Scholar
- [55]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, CA, USA (1987).Google Scholar
- [56]T.L. Magnanti and J.B. Orlin, Parametric linear programming and anti-cycling pivoting rules, Math. Progr. 41(1988)317–325.CrossRefGoogle Scholar
- [57]N. Megiddo, On finding primal- and dual-optimal bases, ORSA J. Comp. 3(1991)63–65.Google Scholar
- [58]N. Megiddo, Switching from a primal-dual Newton algorithm to a primal-dual (interior) simplex-algorithm, RJ 6327 (61996), IBM Almaden Research Center, San Jose, CA, USA (1988).Google Scholar
- [59]K.G. Murty, A note on a Bard type scheme for solving the complementarity problem, Opsearch 11(1974)123–130.Google Scholar
- [60]K.G. Murty,
*Linear and Combinatorial Programming*(Krieger, Malabar, USA, 1976).Google Scholar - [61]K.G. Murty, Computational complexity of parametric linear programming, Math. Progr. 19(1980)213–219.CrossRefGoogle Scholar
- [62]K.G. Murty, The gravitational method of linear programming, Opsearch 23(1986)206–214.Google Scholar
- [63]K.G. Murty,
*Linear Complementarity, Linear and Nonlinear Programming*, Sigma Series in Applied Mathematics, Vol. 3 (Heldermann, Berlin, 1988).Google Scholar - [64]K.G. Murty, A new interior variant of the gradient projection method for linear programming, Technical Report No. 85-18, Department of Industrial and Operations Engineering, The University of Michigan, Ann Arbor, MI 48109-2117, USA (1985).Google Scholar
- [65]M. Namiki and T. Matsui, Some modifications of the criss-cross method, Research Report, Department of Information Sciences, Tokyo Institute of Technology, Tokyo, Japan (1990).Google Scholar
- [66]P.-Q. Pan, Practical finite pivoting rules for the simplex method, OR Spektrum 12(1988)219–225.CrossRefGoogle Scholar
- [67]K. Paparrizos, Pivoting rules directing the simplex method through all feasible vertices of Klee-Minty examples, Opsearch 26(1989)77–95.Google Scholar
- [68]K. Paparrizos, An exterior point simplex algorithm. Lecture at the
*EURO-TIMS Conf.*, Belgrade, Jugoslavia (1989).Google Scholar - [69]K. Paparrizos, A simplex algorithm with a new monotonicity property, Working Paper (1991).Google Scholar
- [70]K. Paparrizos, An infeasible (exterior point) simplex algorithm for assignment problems, Math. Progr. 51(1991)45–54.CrossRefGoogle Scholar
- [71]C. Roos, A pivoting rule for the simplex method which is related to Karmarkar's potential function, Manuscript, Faculty of Technical Mathematics and Informatics, Delft University of Technology, The Netherlands (1986).Google Scholar
- [72]C. Roos, An exponential example for Terlaky's pivoting rule for the criss-cross simplex method, Math. Progr. 46(1990)78–94.CrossRefGoogle Scholar
- [73]D.M. Ryan and M.R. Osborne, On the solution of highly degenerate linear programmes, Math. Progr. 41(1988)385–392.CrossRefGoogle Scholar
- [74]A. Schrijver,
*Theory of Linear and Integer Programming*(J. Wiley, 1987).Google Scholar - [75]A. Sehti and G. Thompson, The pivot-and-probe algorithm for solving a linear problem, Math. Progr. 29(1984)219–233.CrossRefGoogle Scholar
- [76]R.E. Stone and C.A. Tovey, The simplex and projective scaling algorithms as iteratively reweighted least squares methods, SIAM Rev. 33(1991)220–237.CrossRefGoogle Scholar
- [77]A. Tamura, private communication (1991).Google Scholar
- [78]A. Tamura, H. Takehara, K. Fukuda, S. Fujishige and M. Kojima, A dual interior primal simplex method for linear programming. J. Oper. Res. Soc. Japan 31(1988)413–430.Google Scholar
- [79]T. Terlaky, A finite criss-cross method for oriented matroids, J. Combin. Theory (Ser. B) 42(1987)319–327.CrossRefGoogle Scholar
- [80]T. Terlaky, A convergent criss-cross method, Math. Oper. und Stat., ser. Optimization 16(1985)683–690.Google Scholar
- [81]T. Terlaky, A new algorithm for quadratic programming, Eur. J. Oper. Res. 32(1987)294–301.CrossRefGoogle Scholar
- [82]M.J. Todd, Complementarity in oriented matroids, SIAM J. Alg. Discr. Math. 5(1984)467–485.CrossRefGoogle Scholar
- [83]M.J. Todd, Linear and quadratic programming in oriented matroids, J. Combin. Theory (Ser. B) 39(1985)105–133.CrossRefGoogle Scholar
- [84]M.J. Todd, A. Dantzig-Wolfe-like variant of Karmarkar's interior-point linear programming algorithm, Oper. Res. 38(1990)1006–1018.CrossRefGoogle Scholar
- [85]M.J. Todd, Polynomial expected behavior of a pivoting algorithm for linear complementarity and linear programming problems, Math. Progr. 35(1986)173–192.CrossRefGoogle Scholar
- [86]A. Tucker, A note on convergence of the Ford-Fulkerson flow algorithm, Math. Oper. Res. 2(1977)143–144.CrossRefGoogle Scholar
- [87]Zh. Wang, A conformal elimination-free algorithm for oriented matroid programming, Chin. Ann. Math. 8(B1)(1987).Google Scholar
- [88]Zh. Wang, A modified version of the Edmonds-Fukuda algorithm for LP problems in the general form, Asia-Pacific J. Oper. Res. (1989).Google Scholar
- [89]Zh. Wang, A general deterministic pivot method for oriented matroid programming, Chin. Ann. Math. (1990).Google Scholar
- [90]Y. Ye and J.A. Kaliski, Further results on build-up approaches for linear programming, Lecture at the
*ORSA/TIMS Meeting*, Anaheim, CA, USA (1991).Google Scholar - [91]N. Zadeh, What is the worst case behavior of the simplex algorithm?, Technical Report No. 27, Department of Operations Research, Stanford University, Stanford, CA, USA (1980).Google Scholar
- [92]S. Zhang, On anti-cycling pivoting rules for the simplex method, Oper. Res. Lett. 10(1991)189–192.CrossRefGoogle Scholar
- [93]G.M. Ziegler, Linear programming in oriented matroids, Technical Report No. 195, Institut für Mathematik, Universität Augsburg, Germany (1990).Google Scholar
- [94]S. Zionts, The criss-cross method for solving linear programming problems, Manag. Sci. 15(1969)426–445.CrossRefGoogle Scholar