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The Extended Linear Complementarity Problem and Its Applications in Analysis and Control of Discrete-Event Systems

  • Bart De Schutter
Part of the Springer Optimization and Its Applications book series (SOIA, volume 17)

In this chapter, we give an overview of complementarity problems with a special focus on the extended linear complementarity problem (ELCP) and its applications in analysis and control of discrete-event systems such as traffic signal controlled intersections, manufacturing systems, railway networks, etc. We start by giving an introduction to the (regular) linear complementarity problem (LCP). Next, we discuss some extensions, with a particular emphasis on the ELCP, which can be considered to be the most general linear extension of the LCP. We then discuss some algorithms to compute one or all solutions of an ELCP. Next, we present a link between the ELCP and max-plus equations. This is then the basis for some applications of the ELCP in analysis and model-based predictive control of a special class of discrete-event systems. We also show that — although the general ELCP is NP-hard — the ELCP-based control problem can be transformed into a linear programming problem, which can be solved in polynomial time.

Keywords

linear complementarity problem extended linear complementarity problem algorithms control applications discrete-event systems maxplus-linear systems 

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References

  1. 1.
    Andreani, R., Martínez, J.M.: On the Solution of the Extended Linear Complementarity Problem. Linear Algebra and Its Applications, 281, 247–257 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Atamtürk, A., Savelsbergh, M.W.P.: Integer-Programming Software Systems. Annals of Operations Research, 140(1), 67–124 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Baccelli, F., Cohen, G., Olsder, G.J., Quadrat, J.P.: Synchronization and Linearity. John Wiley & Sons, New York (1992)zbMATHGoogle Scholar
  4. 4.
    Bai, Z.Z.: On the Convergence of the Multisplitting Methods for the Linear Complementarity Problem. SIAM Journal on Matrix Analysis and Applications, 21(1), 67–78 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Bemporad, A., Morari, M.: Control of Systems Integrating Logic, Dynamics, and Constraints. Automatica, 35(3), 407–427 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Chen, C., Mangasarian, O.L.: Smoothing Methods for Convex Inequalities and Linear Complementarity Problems. Mathematical Programming, 71(1), 51–69 (1995)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Chung, S.: NP-Completeness of the Linear Complementarity Problem. Journal of Optimization Theory and Applications, 60(3), 393–399 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Cordier, C., Marchand, H., Laundy, R., Wolsey, L.A.: bc-opt: A branch-and-Cut Code for Mixed Integer Programs. Mathematical Programming, Series A, 86(2), 335–353 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Cottle, R.W., Dantzig, G.B.: A Generalization of the Linear Complementarity Problem. Journal of Combinatorial Theory, 8(1), 79–90 (1970)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Cottle, R.W., Pang, J.S., Stone, R.E.: The Linear Complementarity Problem. Academic Press, Boston (1992)zbMATHGoogle Scholar
  11. 11.
    Cuninghame-Green, R.A.: Minimax Algebra. Lecture Notes in Economics and Mathematical Systems. Vol 166, Springer-Verlag, Berlin, Germany (1979)Google Scholar
  12. 12.
    Cutler, C.R., Ramaker, B.L.: Dynamic Matrix Control. A Computer Control Algorithm. In: Proceedings of the 86th AIChE National Meeting, Houston, Texas (1979)Google Scholar
  13. 13.
    De Moor, B., Vandenberghe, L., Vandewalle, J.: The Generalized Linear Complementarity Problem and an Algorithm to Find All Its Solutions. Mathematical Programming, 57, 415–426 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    De Schutter, B.: Max-Algebraic System Theory for Discrete Event Systems. PhD thesis, Faculty of Applied Sciences, K.U. Leuven, Leuven, Belgium (1996)Google Scholar
  15. 15.
    De Schutter, B.: Optimal Control of a Class of Linear Hybrid Systems with Saturation. SIAM Journal on Control and Optimization, 39(3), 835–851 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    De Schutter, B.: Optimizing Acyclic Traffic Signal Switching Sequences Through an Extended Linear Complementarity Problem Formulation. European Journal of Operational Research, 139(2), 400–415 (2002)zbMATHCrossRefGoogle Scholar
  17. 17.
    De Schutter, B., De Moor, B.: The Extended Linear Complementarity Problem. Mathematical Programming, 71(3), 289–325 (1995)CrossRefMathSciNetGoogle Scholar
  18. 18.
    De Schutter, B., De Moor, B.: Minimal Realization in the Max Algebra is an Extended Linear Complementarity Problem. Systems & Control Letters, 25(2), 103–111 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    De Schutter, B., De Moor, B.: A Method to Find All Solutions of a System of Multivariate Polynomial Equalities and Inequalities in the Max Algebra. Discrete Event Dynamic Systems: Theory and Applications, 6(2), 115–138 (1996)zbMATHCrossRefGoogle Scholar
  20. 20.
    De Schutter, B., De Moor, B.: The Linear Dynamic Complementarity Problem is a Special Case of the Extended Linear Complementarity Problem. Systems & Control Letters, 34, 63–75 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    De Schutter, B., De Moor, B.: Optimal Traffic Light Control for a Single Intersection. European Journal of Control, 4(3), 260–276 (1998)zbMATHGoogle Scholar
  22. 22.
    De Schutter, B., De Moor, B.: The QR Decomposition and the Singular Value Decomposition in the Symmetrized Max-Plus Algebra. SIAM Journal on Matrix Analysis and Applications, 19(2), 378–406 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    De Schutter, B., Heemels, W.P.M.H., Bemporad, A.: On the Equivalence of Linear Complementarity Problems. Operations Research Letters, 30(4), 211–222 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    De Schutter, B., van den Boom, T.: Model Predictive Control for Max-Min-Plus Systems. In Boel, R. and Stremersch, G., editors, Discrete Event Systems: Analysis and Control, Kluwer International Series in Engineering and Computer Science, Vol. 569, 201–208. Kluwer Academic Publishers, Boston (2000)Google Scholar
  25. 25.
    De Schutter, B., van den Boom, T.: Model Predictive Control for Max-Plus-Linear Discrete Event Systems. Automatica, 37(7), 1049–1056 (2001)zbMATHCrossRefGoogle Scholar
  26. 26.
    De Schutter, B., van den Boom, T.J.J.: Model Predictive Control for Max-Min-Plus-Scaling Systems. In Proceedings of the 2001 American Control Conference, 319–324, Arlington, Virginia (2001)Google Scholar
  27. 27.
    Eaves, B.C.: The Linear Complementarity Problem. Management Science, 17(9), 612–634 (1971)zbMATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Ebiefung, A.A., Kostreva, M.K.: Global Solvability of Generalized Linear Complementarity Problems and a Related Class of Polynomial Complementarity Problems. In: Floudas, C.A., Pardalos, P.M. (eds) Recent Advances in Global Optimization, Princeton Series in Computer Science, 102–124. Princeton University Press, Princeton, New Jersey (1992)Google Scholar
  29. 29.
    Ferris, M.C., Mangasarian, O.L., Pang, J.S. (eds): Complementarity: Applications, Algorithms and Extensions, Applied Optimization, Vol. 50, Springer, New York, (2001)Google Scholar
  30. 30.
    Ferris, M.C., Pang, J.S. (eds): Complementarity and Variational Problems: State of the Art. Philadelphia, Pennsylvania: SIAM. Proceedings of the International Conference on Complementarity Problems, Baltimore, Maryland, November 1995, (1997)Google Scholar
  31. 31.
    Ferris, M.C., Pang, J.S.: Engineering and Economic Applications of Complementarity Problems. SIAM Review, 39(4), 669–713 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    Fletcher, R., Leyffer, S.: Numerical Experience with Lower Bounds for MIQP Branch-and-Bound. SIAM Journal on Optimization, 8(2), 604–616 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    Gowda, M.S.: On the Extended Linear Complementarity Problem. Mathematical Programming, 72, 33–50 (1996)MathSciNetGoogle Scholar
  34. 34.
    Gowda, M.S., Sznajder, R.: The Generalized Order Linear Complementarity Problem. SIAM Journal on Matrix Analysis and Applications, 15(3), 779–795 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  35. 35.
    Heemels, W.P.M.H., De Schutter, B., Bemporad, A.: Equivalence of Hybrid Dynamical Models. Automatica, 37(7), 1085–1091 (2001)zbMATHCrossRefGoogle Scholar
  36. 36.
    Heemels, W.P.M.H., Schumacher, J.M., Weiland, S.: Linear Complementarity Systems. SIAM Journal on Applied Mathematics, 60(4), 1234–1269 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  37. 37.
    Isac, G.: Complementarity Problems. Springer-Verlag, Berlin, Germany (1992)zbMATHGoogle Scholar
  38. 38.
    Isac, G., Bulavsky, V.A., Kalashnikov, V.V.: Complementarity, Equilibrium, Efficiency and Economics, Nonconvex Optimization and Its Applications, Vol. 63, Springer (2002)Google Scholar
  39. 39.
    Júdice, J.J., Vicente, L.N.: On the Solution and Complexity of a Generalized Linear Complementarity Problem. Journal of Global Optimization, 4(4), 415–424 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  40. 40.
    Kaliski, J.A., Ye, Y.: An Extension of the Potential Reduction Algorithm for Linear Complementarity Problems with Some Priority Goals. Linear Algebra and Its Applications, 193, 35–50 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  41. 41.
    Kanzow, C.: Some Noninterior Continuation Methods for Linear Complementarity Problems. SIAM Journal on Matrix Analysis and Applications, 17(4), 851–868 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  42. 42.
    Kočvara, M., Zowe, J.: An Iterative Two-Step Algorithm for Linear Complementarity Problems. Numerische Mathematik, 68(1), 95–106 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  43. 43.
    Kremers, H., Talman, D.: A New Pivoting Algorithm for the Linear Complementarity Problem Allowing for an Arbitrary Starting Point. Mathematical Programming, 63(2), 235–252 (1994)CrossRefMathSciNetGoogle Scholar
  44. 44.
    Li, T.Y.: Numerical Solution of Polynomial Systems by Homotopy Continuation Methods. In: Cucker, F. (ed) Handbook of Numerical Analysis, Vol. XI, Special Volume: Foundations of Computational Mathematics, 209–304. North-Holland, Amsterdam (2003)Google Scholar
  45. 45.
    Linderoth, J., Ralphs, T.: Noncommercial Software for Mixed-Integer Linear Programming. Optimization Online. (2004) See http://www.optimization-online.org/DBHTML/2004/12/1028.html.
  46. 46.
    Maciejowski, J.M.: Predictive Control with Constraints. Prentice Hall, Harlow, England (2002)Google Scholar
  47. 47.
    Mangasarian, O.L. The Linear Complementarity Problem as a Separable Bilinear Program. Journal of Global Optimization, 6(2), 153–161 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  48. 48.
    Mangasarian, O.L., Pang, J.S.: The Extended Linear Complementarity Problem. SIAM Journal on Matrix Analysis and Applications, 16(2), 359–368 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  49. 49.
    Mangasarian, O.L., Solodov, M.V.: Nonlinear Complementarity as Unconstrained and Constrained Minimization. Mathematical Programming, 62(2), 277–297 (1993)CrossRefMathSciNetGoogle Scholar
  50. 50.
    McShane, K.: Superlinearly Convergent \(O(\sqrt{n}L)\)-Iteration Interior-Point Algorithms for Linear Programming and the Monotone Linear Complementarity Problem. SIAM Journal on Optimization, 4(2), 247–261 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  51. 51.
    Mehrotra, S., Stubbs, R.A.: Predictor-Corrector Methods for a Class of Linear Complementarity Problems. SIAM Journal on Optimization, 4(2), 441–453 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  52. 52.
    Mohan, S.R., Neogy, S.K., Sridhar, R.: The Generalized Linear Complementarity Problem Revisited. Mathematical Programming, 74, 197–218 (1996)MathSciNetGoogle Scholar
  53. 53.
    Motzkin, T.S., Raiffa, H., Thompson, G.L., Thrall, R.M.: The Double Description Method. In: Kuhn, H.W., Tucker, A.W. (eds) Contributions to the Theory of Games, Annals of Mathematics Studies, number 28, 51–73. Princeton University Press, Princeton, New Jersey (1953)Google Scholar
  54. 54.
    Murty, K.G.: Linear Complementarity, Linear and Nonlinear Programming. Helderman Verlag, Berlin, Germany (1988)zbMATHGoogle Scholar
  55. 55.
    Nesterov, Y., Nemirovskii, A.: Interior-Point Polynomial Algorithms in Convex Programming. SIAM, Philadelphia, Pennsylvania (1994)zbMATHGoogle Scholar
  56. 56.
    Pardalos, P.M., Resende, M.G.C. (eds): Handbook of Applied Optimization. Oxford University Press, Oxford, UK (2002)zbMATHGoogle Scholar
  57. 57.
    Richalet, J., Rault, A., Testud, J.L., Papon, J.: Model Predictive Heuristic Control: Applications to Industrial Processes. Automatica, 14(5), 413–428 (1978)CrossRefGoogle Scholar
  58. 58.
    Schäfer, U.: On the Modulus Algorithm for the Linear Complementarity Problem. Operations Research Letters, 32(4), 350–354 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  59. 59.
    Schumacher, J.M.: Some Modeling Aspects of Unilaterally Constrained Dynamics. In: Proceedings of the ESA International Workshop on Advanced Mathematical Methods in the Dynamics of Flexible Bodies, ESA-ESTEC, Noordwijk, The Netherlands (1996)Google Scholar
  60. 60.
    Sheng, R., Potra, F.A.: A Quadratically Convergent Infeasible-Interiorpoint Algorithm for LCP with Polynomial Complexity. SIAM Journal on Optimization, 7(2), 304–317 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  61. 61.
    Sontag, E.D.: Nonlinear Regulation: The Piecewise Linear Approach. IEEE Transactions on Automatic Control, 26(2), 346–358 (1981)zbMATHCrossRefMathSciNetGoogle Scholar
  62. 62.
    Sznajder, R., Gowda, M.S.: Generalizations of P 0- and P-Properties; Extended Vertical and Horizontal Linear Complementarity Problems. Linear Algebra and Its Applications, 223/224, 695–715 (1995)Google Scholar
  63. 63.
    Taha, H.A.: Operations Research: An Introduction. 4th edition, Macmillan Publishing Company, New York (1987)Google Scholar
  64. 64.
    Vandenberghe, L., De Moor, B., Vandewalle, J.: The Generalized Linear Complementarity Problem Applied to the Complete Analysis of Resistive Piecewise-Linear Circuits. IEEE Transactions on Circuits and Systems, 36(11), 1382–1391 (1989)CrossRefGoogle Scholar
  65. 65.
    Wright, S.J.: An Infeasible-Interior-Point Algorithm for Linear Complementarity Problems. Mathematical Programming, 67(1), 29–51 (1994)CrossRefMathSciNetGoogle Scholar
  66. 66.
    Ye, Y.: A Fully Polynomial-Time Approximation Algorithm for Computing a Stationary Point of the General Linear Complementarity Problem. Mathematics of Operations Research, 18(2), 334–345 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  67. 67.
    Yuan, D., Song, Y.: Modified AOR Methods for Linear Complementarity Problem. Applied Mathematics and Computation, 140(1), 53–67 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  68. 68.
    Zhang, Y.: On the Convergence of a Class of Infeasible Interior-Point Methods for the Horizontal Linear Complementarity Problem. SIAM Journal on Optimization, 4(1), 208–227 (1994)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  • Bart De Schutter
    • 1
  1. 1.Delft Center for Systems and ControlDelft University of TechnologyNetherlands

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