# Some generalizations of the criss-cross method for the linear complementarity problem of oriented matroids

- 63 Downloads
- 9 Citations

## Abstract

Quadratic programming, symmetry, positive (semi) definiteness and the linear complementary problem were generalized by Morris and Todd to oriented matroids. Todd gave a constructive solution for the quadratic programming problem of oriented matroids. Using Las Vergnas' lexicographic extension and Bland's basic tableau construction Todd generalized Lemke's quadratic programming algorithm for this problem.

Here some generalizations of Terlaky's finite criss-cross method are presented for oriented matroid quadratic programming. These algorithms are based on the smallest subscript rule and on sign patterns, and do not preserve feasibility on any subsets. In fact two variants of the generalized criss-cross method are presented. Finally two special cases (oriented matroid linear programming and the definite case) are discussed.

## AMS subject classification (1980)

05 B 35 90 C 48## Preview

Unable to display preview. Download preview PDF.

## References

- [1]M. L. Balinski andA. W. Tucker, Duality theory of linear programs: A constructive approach with applications,
*SIAM Review*,**11**(1969), 347–377.Google Scholar - [2]E. M. L. Beale, On quadratic programming,
*Naval Research Logistics Quarterly*,**6**(1959), 227–244.Google Scholar - [3]M. J. Best, Equivalence of some quadratic programming algorithms,
*Mathematical Programming*,**30**(1984), 71–87.Google Scholar - [4]R. G. Bland, New finite pivoting rules for the simplex method,
*Mathematics of Operations Research*,**2**(1977), 103–107.Google Scholar - [5]R. G. Bland, A combinatorial abstraction of linear programming,
*J. Combinatorial Theory Ser. B.*,**23**(1977), 33–57.Google Scholar - [6]R. G. Bland andM. Las Vergnas, Orientability of matroids,
*J. Combinatorial Theory Ser. B.*,**24**(1978), 94–123.Google Scholar - [7]Y. Y. Chang andR. W. Cottle, Least index resolution of degeneracy in quadratic programming,
*Mathematical Programming*,**18**(1980), 127–137.Google Scholar - [8]R. W. Cottle, The principal pivoting method of quadratic programming, in:
*Mathematics of the Decision Sciences Part I*. Eds. G. B. Dantzig and A. F. Veinott, Lectures in Applied Mathematics, 11 (American Mathematical Society, Providence, R. I., (1968), 142–162.Google Scholar - [9]R. W. Cottle andG. B. Dantzig, Complementary pivot theory of mathematical programming,
*Linear Algebra and its Applications*,**1**(1986), 103–125.Google Scholar - [10]G. B. Dantzig,
*Linear programming and Extensions*, Princeton University Press, Princeton, N. J., 1963.Google Scholar - [11]J. Folkman andJ. Lawrence, Oriented matroids,
*J. Combinatorial Theory Ser. B.*,**25**(1978), 199–236.Google Scholar - [12]K.Fukuda, Oriented matroid programming,
*Ph. D. dissertation, University of Waterloo*, 1982.Google Scholar - [13]D.Jensen, Coloring and Duality: Combinatorial Augmentation methods,
*Ph. D. dissertation School of OR and IE, Cornell University*, 1985.Google Scholar - [14]E. Keller, The general quadratic optimization problem,
*Mathematical Programming*,**5**(1973), 311–337.Google Scholar - [15]E.Klafszky and T.Terlaky, Variants of the „Hungarian Method” for solving linear programming problems,
*Optimization (accepted for publication)*.Google Scholar - [16]E.Klafszky and T.Terlaky, Some generalizations of the criss-cross method for quadratic programming,
*Mathematics of Operations Research (to appear)*.Google Scholar - [17]M. Las Vergnas, Bases in oriented matroids,
*J. Combinatorial Theory Ser. B.*,**25**(1978), 283–289.Google Scholar - [18]M. Las Vergnas, Convexity in oriented matroids,
*J. Combinatorial Theory Ser. B.*,**29**(1980), 231–243.Google Scholar - [19]J. Lawrence, Oriented matroids and multiply ordered sets,
*Linear Algebra and its Applications*,**48**(1982), 1–12.Google Scholar - [20]C. E. Lemke, Bimatrix equilibrium points and mathematical programming,
*Management Science*,**11**(1965), 681–689.Google Scholar - [21]W. D.Morris, Oriented matroids and the linear complementarity problem,
*Ph. D. thesis. School of OR and IE, Cornell University*, 1986.Google Scholar - [22]W. D.Morris and M. J.Todd, Symmetry and positive definiteness in oriented matroids,
*Cornell University, School of OR and IE, Itacha, Itacha N. Y. Technical Report*No.**631**.Google Scholar - [23]R. T. Rockafellar, The elementary vectors of a subspace of
*R*^{n}, in*Combinatorial Mathematics and its Applications*, Proc. of the Chapel Hill Conference, 1967 (R. G. Bore and T. A. Dowling, Eds.) 104–127, Univ. of North Carolina Press, Chapel Hill. 1969.Google Scholar - [24]K.Ritter, A dual quadratic programming algorithm,
*Univ. of Wisconsin-Madison, Mathematics Research Center, Technical Summary Report*, No. 2733 (1984).Google Scholar - [25]C.Roos, On the Terlaky path in the umbrella graph of a linear programming problem,
*Reports of the Department of Informatics, Delft University of Technology*, No. 85-12.Google Scholar - [26]
- [27]T. Terlaky, A finite criss-cross method for oriented matroids,
*J. Combinatorial Theory Ser. B.*,**42**(1987), 319–327.Google Scholar - [28]T. Terlaky, A new algorithm for quadratic programming,
*European Journal of Operations Research*,**32**(1987), 294–301.Google Scholar - [29]T. Terlaky, On
*lp*programming,*European Journal of Operations Research*,**22**(1985), 70–101.Google Scholar - [30]M. J. Todd, Complementarity in oriented matroids,
*SIAM J. Alg. Disc. Math.*,**5**(1984), 467–485.Google Scholar - [31]M. J. Todd, Linear and quadratic programming in oriented matroids,
*J. Combinatorial Theory Ser. B.*,**39**(1985), 105–133.Google Scholar - [32]A. W. Tucker, Principal pivotal transforms of square matrices,
*SIAM Review*,**5**(1963), 305.Google Scholar - [33]C. Van de Panne andA. Whinston, The symmetric formulation of the simplex method for quadratic programming,
*Econometrica*,**37**(1969), 507–527.Google Scholar - [34]
- [35]P. Wolfe, The simplex method for quadratic programming,
*Econometrica*,**27**(1959), 382–398.Google Scholar - [36]Zh. Wang, A finite conformal-elimination-free algorithm for oriented matroid programming,
*Chinecse Annals of Mathematics*,**8B**. No. 1 (1987).Google Scholar - [37]H. Whitney, On the abstract properties of linear dependence,
*Amer. J. Math.*,**57**(1935), 507–553.Google Scholar - [38]S. Zionts, The criss-cross method for solving linear programming problems,
*Management Science*,**15**(1969), 426–445.Google Scholar