Annals of Operations Research

, Volume 274, Issue 1–2, pp 531–553

# Total dual integrality of the linear complementarity problem

Original Research

## Abstract

In this paper, we introduce total dual integrality of the linear complementarity problem (LCP) by analogy with the linear programming problem. The main idea of defining the notion is to propose the LCP with orientation, a variant of the LCP whose feasible complementary cones are specified by an additional input vector. Then we naturally define the dual problem of the LCP with orientation and total dual integrality of the LCP. We show that if the LCP is totally dual integral, then all basic solutions are integral. If the input matrix is sufficient or rank-symmetric, and the LCP is totally dual integral, then our result implies that the LCP always has an integral solution whenever it has a solution. We also introduce a class of matrices such that any LCP instance having the matrix as a coefficient matrix is totally dual integral. We investigate relationships between matrix classes in the LCP literature such as principally unimodular matrices. Principally unimodular matrices are known that all basic solutions to the LCP are integral for any integral input vector. In addition, we show that it is coNP-hard to decide whether a given LCP instance is totally dual integral.

## Keywords

Linear complementarity problem Total dual integrality Principal unimodularity

## References

1. Bouchet, A. (1992). A characterization of unimodular orientations of simple graphs. Journal of Combinatorial Theory, Series B, 56(1), 45–54.
2. Chandrasekaran, R. (1984). Integer programming problems for which a simple rounding type algorithm works. Progress in Combinatorial Optimization, 8, 101–106.
3. Chandrasekaran, R., Kabadi, S. N., & Sridhar, R. (1998). Integer solution for linear complementarity problem. Mathematics of Operations Research, 23(2), 390–402.
4. Chung, S. J. (1989). NP-completeness of the linear complementarity problem. Journal of Optimization Theory and Applications, 60(3), 393–399.
5. Cook, W., Lovász, L., & Schrijver, A. (1984). A polynomial-time test for total dual integrality in fixed dimension (pp. 64–69). Berlin Heidelberg, Berlin, Heidelberg: Springer.Google Scholar
6. Cottle, R. W. (1968). The principal pivoting method of quadratic programming. In Mathematics of decision sciences, part 1 (pp. 142–162). Providence R.I.: American Mathematical Society.Google Scholar
7. Cottle, R. W., & Dantzig, G. B. (1968). Complementary pivot theory of mathematical programming. Linear Algebra and its Applications, 1(1), 103–125.
8. Cottle, R. W., Pang, J. S., & Stone, R. E. (1992). The linear complementarity problem. Boston: Academic Press.Google Scholar
9. Cottle, R. W., Pang, J. S., & Venkateswaran, V. (1989). Sufficient matrices and the linear complementarity problem. Linear Algebra and its Applications, 114–115, 231–249 (special Issue Dedicated to Alan J. Hoffman).Google Scholar
10. Cunningham, W. H., & Geelen, J. F. (1998). Integral solutions of linear complementarity problems. Mathematics of Operations Research, 23(1), 61–68.
11. Ding, G., Feng, L., & Zang, W. (2008). The complexity of recognizing linear systems with certain integrality properties. Mathematical Programming, 114(2), 321–334.
12. Edmonds, J., & Giles, R. (1977). A min-max relation for submodular functions on graphs. In P. L. Hammer, E. L. Johnson, B. H. Korte & G. L. Nemhauser (Eds.), Studies in integer programming, Annals of discrete mathematics (Vol. 1, pp. 185–204). Amsterdam: Elsevier.Google Scholar
13. Fukuda, K., & Terlaky, T. (1992). Linear complementarity and oriented matroids. Journal of the Operations Research Society of Japan, 35(1), 45–61.
14. Gabriel, S. A., Conejo, A. J., Ruiz, C., & Siddiqui, S. (2013). Solving discretely constrained, mixed linear complementarity problems with applications in energy. Computers & Operations Research, 40(5), 1339–1350.
15. Garey, M. R., & Johnson, D. S. (1990). Computers and intractability; A guide to the theory of NP-completeness. New York: W. H. Freeman & Co.Google Scholar
16. Harville, D. A. (1997). Matrix algebra from a statistician’s perspective. New York: Springer.Google Scholar
17. Hoffman, A. J., & Kruskal, J. B. (1956). Integral boundary points of convex polyhedra. In H. Kuhn & A. Tucker (Eds.), Linear inequalities and related systems (pp. 223–246). Princeton: Princeton University Press.Google Scholar
18. Howson, J. T., Jr. (1972). Equilibria of polymatrix games. Management Science, 18(5-part-1), 312–318Google Scholar
19. Kronecker, L. (1884). Näherungsweise ganzzahlige auflösung linearer gleichungen. Monatsberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin, 1179–1193, 1271–1299.Google Scholar
20. Lemke, C. E. (1965). Bimatrix equilibrium points and mathematical programming. Management Science, 11(7), 681–689.
21. Murty, K. G. (1997). Linear complementarity, linear and nonlinear programming. Internet Edition, http://www-personal.umich.edu/~murty/books/linear_complementarity_webbook/
22. Pap, J. (2011). Recognizing conic TDI systems is hard. Mathematical Programming, 128(1), 43–48.
23. Pardalos, P. M., & Nagurney, A. (1990). The integer linear complementarity problem. International Journal of Computer Mathematics, 31(3–4), 205–214.
24. Ruiz, C., Conejo, A. J., & Gabriel, S. A. (2012). Pricing non-convexities in an electricity pool. IEEE Transactions on Power Systems, 27(3), 1334–1342.
25. Schrijver, A. (1986). Theory of linear and integer programming. Toronto: WileyGoogle Scholar
26. Takayama, T., & Judge, G. G. (1971). Spatial and temporal price allocation models. Amsterdam: North-Holland.Google Scholar

© Springer Science+Business Media, LLC, part of Springer Nature 2018

## Authors and Affiliations

• Hanna Sumita
• 1
Email author
• Naonori Kakimura
• 2
• Kazuhisa Makino
• 3
1. 1.Faculty of Economics and Business AdministrationTokyo Metropolitan UniversityTokyoJapan
2. 2.Department of MathematicsKeio UniversityYokohamaJapan
3. 3.Research Institute for Mathematical SciencesKyoto UniversityKyotoJapan

## Personalised recommendations 