Mathematical Programming

, Volume 101, Issue 1, pp 263–295 | Cite as

Complementarity systems in optimization

Article

Abstract.

Complementarity systems consist of ordinary differential equations coupled to complementarity conditions. They form a class of nonsmooth dynamical systems that is of use in mechanical and electrical engineering as well as in optimization and in other fields. The paper illustrates how complementarity systems arise in mathematical programming by means of a number of examples of various nature. This is followed by a brief survey of the results that are available concerning existence, uniqueness, and generation of solutions. The emphasis in this paper is on linear complementarity systems.

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References

  1. 1.
    Anitescu, M., Potra, F.A.: Formulating dynamic multi-rigid-body contact problems with friction as solvable Linear Complementarity Problems. Nonlin. Dyn. 14, 231–247 (1997)CrossRefMathSciNetMATHGoogle Scholar
  2. 2.
    Aubin, J.P., Cellina, A.: Differential Inclusions. Springer, Berlin, 1984Google Scholar
  3. 3.
    Baccelli, F., Cohen, G.,Olsder, G.J., Quadrat, J.-P.: Synchronization and Linearity. An Algebra for Discrete Event Systems. Wiley, New York, 1992Google Scholar
  4. 4.
    Berman, A., Plemmons, R.J.: Nonnegative Matrices in the Mathematical Sciences. Academic Press, New York, 1979Google Scholar
  5. 5.
    Berridge, S.J., Schumacher, J.M.: An irregular grid method for high-dimensional free-boundary problems in finance. Future Generation Comput. Syst. 20, 353–362 (2004)CrossRefGoogle Scholar
  6. 6.
    van Bokhoven, W.M.G.: Piecewise Linear Modelling and Analysis. Kluwer, Deventer, the Netherlands, 1981Google Scholar
  7. 7.
    Brenan, K.E., Campbell, S.L., Petzold, L.R.: Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations. Classics in Applied Mathematics, 14. SIAM, Philadelphia, 1996. Extended reprint of original edition, 1989Google Scholar
  8. 8.
    Brockett, R.W.: Smooth multimode control systems. In: Hunt, L., Martin, C. (eds.), Proc. 1983 Berkeley-Ames Conference on Nonlinear Problems in Control and Fluid Mechanics, Math. Sci. Press, Brookline, 1984, pp. 103–110Google Scholar
  9. 9.
    Çamlibel, M.K., Heemels, W.P.M.H., Schumacher, J.M.: Consistency of a time-stepping method for a class of piecewise-linear networks. IEEE Trans. Circuits Syst. I 49, 349–357 (2002)CrossRefMathSciNetGoogle Scholar
  10. 10.
    Çamlibel, M.K., Heemels, W.P.M.H., van der Schaft, A.J., Schumacher, J.M.: Switched networks and complementarity. IEEE Trans. Circuits Syst. I 50, 1036–1046 (2003)CrossRefMathSciNetGoogle Scholar
  11. 11.
    Çamlibel,M.K., Schumacher, J.M.: On the Zeno behavior of linear complementarity systems. In: Proc. 40th IEEE Conf. Dec. Contr. (Orlando, Fl., Dec. 2001), 2001, pp. 346–351Google Scholar
  12. 12.
    Chen, H., Mandelbaum, A.: Leontief systems, RBV’s and RBM’s. In: Davis, M.H.A., Elliott, R.J. (eds.), Applied Stochastic Analysis, Stochastics Monographs, vol. 5, Gordon & Breach, New York, 1991, pp. 1–43Google Scholar
  13. 13.
    Curtain, R.F., Zwart, H.J.: An Introduction to Infinite-Dimensional Linear Systems Theory. Springer, New York, 1995Google Scholar
  14. 14.
    Dupuis, P., Nagurney, A.: Dynamical systems and variational inequalities. Ann. Oper. Res. 44, 9–42 (1993)MathSciNetMATHGoogle Scholar
  15. 15.
    Duvaut, G., Lions, J.L.: Inequalities in Mechanics and Physics. Grundlehren der mathematischen Wissenschaften 219. Springer, Berlin, 1976. Translation of French original, 1972Google Scholar
  16. 16.
    Brogliato, B.: Impacts in Mechanical Systems. Analysis and Modelling. Lecture Notes in Physics, vol. 551. Springer, Berlin, 2000Google Scholar
  17. 17.
    Facchinei, F., Pang, J.-S.: Finite-Dimensional Variational Inequalities and Complementarity Problems (Vol. I, II). Springer, New York, 2003Google Scholar
  18. 18.
    Ferris, M.C., Pang, J.-S.: Engineering and economic applications of complementarity problems. SIAM Rev 39, 669–713 (1997)CrossRefMathSciNetMATHGoogle Scholar
  19. 19.
    Filippov, A.F.: Differential Equations with Discontinuous Righthand Sides. Kluwer, Dordrecht, 1988Google Scholar
  20. 20.
    Gantmacher, F.R.: The Theory of Matrices (Vol. II). Chelsea, New York, 1959Google Scholar
  21. 21.
    Glowinski, R., Lions, J.L., Trémolières, R.: Numerical Analysis of Variational Inequalities. North-Holland, Amsterdam, 1981. Translation of French original, 1976Google Scholar
  22. 22.
    Harrison, J.M.: Brownian Motion and Stochastic Flow Systems. Wiley, New York, 1985Google Scholar
  23. 23.
    Harrison, J.M., Reiman, M.I.: Reflected Brownian motion on an orthant. Ann. Probability 9, 302–308 (1981)MathSciNetMATHGoogle Scholar
  24. 24.
    Hartl, R.F., Sethi, S.P., Vickson, R.G.: A survey of the maximum principles for optimal control problems with state constraints. SIAM Rev. 37, 181–218 (1995)MathSciNetMATHGoogle Scholar
  25. 25.
    Heemels, W.P.M.H., Çamlibel, M.K., Schumacher, J.M.: On the dynamic analysis of piecewise-linear networks. IEEE Trans. Circuits Syst. I 49, 315–327 (2002)CrossRefMathSciNetGoogle Scholar
  26. 26.
    Heemels, W.P.M.H., Çamlibel, M.K., van der Schaft, A.J., Schumacher, J.M.: Modelling, well-posedness, and stability of switched electrical networks. In: Maler, O., Pnueli, A. (eds.), Hybrid Systems: Computation and Control (Proc. 6th Int. Workshop, HSCC03, Prague, April 2003), Springer, Berlin, 2003, pp. 249–266Google Scholar
  27. 27.
    Heemels, W.P.M.H., De Schutter, B., Bemporad, A.: Equivalence of hybrid dynamical models. Automatica 37 (7), 1085–1091 (2001)CrossRefMATHGoogle Scholar
  28. 28.
    Heemels, W.P.M.H., Schumacher, J.M., Weiland, S.: The rational complementarity problem. Lin. Alg. Appl. 294, 93–135 (1999)CrossRefMathSciNetMATHGoogle Scholar
  29. 29.
    Heemels, W.P.M.H., Schumacher, J.M., Weiland, S.: Linear complementarity systems. SIAM J. Appl. Math. 60, 1234–1269 (2000)CrossRefMathSciNetMATHGoogle Scholar
  30. 30.
    Heemels, W.P.M.H., Schumacher, J.M., Weiland, S.: Projected dynamical systems in a complementarity formalism. Oper. Res. Lett. 27, 83–91 (2000)CrossRefMathSciNetMATHGoogle Scholar
  31. 31.
    Kailath, T.: Linear Systems. Prentice-Hall, Englewood Cliffs, N.J., 1980Google Scholar
  32. 32.
    Karamardian, S.: Generalized complementarity problem. J. Optim. Theor. Appl. 8, 161–167 (1971)MATHGoogle Scholar
  33. 33.
    Kilmister, C.W., Reeve, J.E.: Rational Mechanics. Longmans, London, 1966Google Scholar
  34. 34.
    Kuijper, M., Schumacher, J.M.: Input/output structure of differential-algebraic systems. IEEE Trans. Automat. Contr. AC-38, 404–414 (1993)Google Scholar
  35. 35.
    Lions, J.L., Stampacchia, G.: Variational inequalities. Commun. Pure Appl. Math. 20, 493–519 (1967)MATHGoogle Scholar
  36. 36.
    Lötstedt, P.: Coulomb friction in two-dimensional rigid body systems. Zeitschrift für Angewandte Mathematik und Mechanik 61, 605–615 (1981)MathSciNetGoogle Scholar
  37. 37.
    Lötstedt, P.: Mechanical systems of rigid bodies subject to unilateral constraints. SIAM J. Appl. Math. 42, 281–296 (1982)MathSciNetGoogle Scholar
  38. 38.
    Moreau, J.J.: Liaisons unilatérales sans frottement et chocs inélastiques. C. R. Acad. Sc. Paris 296, 1473–1476 (1983)MathSciNetMATHGoogle Scholar
  39. 39.
    Nijmeijer, H., Schumacher, J.M.: On the inherent integration structure of nonlinear systems. IMA J. Math. Contr. Inf. 2, 87–107 (1985)MATHGoogle Scholar
  40. 40.
    Nijmeijer, H., van der Schaft, A.J.: Nonlinear Dynamical Control Systems. Springer-Verlag, Berlin, 1990Google Scholar
  41. 41.
    Pang, J.-S., Stewart, D.E.: Differential variational inequalities. Preprint, 2003Google Scholar
  42. 42.
    Pérès, J.: Mécanique Générale. Masson & Cie., Paris, 1953Google Scholar
  43. 43.
    Pfeiffer, F., Glocker, C.: Multibody Dynamics with Unilateral Contact. Wiley, New York, 1996Google Scholar
  44. 44.
    Pogromsky, A. Yu., Heemels, W.P.M.H., Nijmeijer, H.: On solution concepts and well-posedness of linear relay systems. Automatica 39, 2139–2147 (2003)CrossRefMATHGoogle Scholar
  45. 45.
    Pontryagin, L.S., Boltyanskii, V.G., Gamkrelidze, R.V., Mishchenko, E.F.: The Mathematical Theory of Optimal Processes. Interscience, New York, 1962Google Scholar
  46. 46.
    Reiman, M.I.: Open queueing networks in heavy traffic. Math. Oper. Res. 9, 441–458 (1984)MathSciNetMATHGoogle Scholar
  47. 47.
    van der Schaft, A.J., Schumacher, J.M.: The complementary-slackness class of hybrid systems. Math. Control, Signals Syst. 9, 266–301 (1996)Google Scholar
  48. 48.
    van der Schaft, A.J., Schumacher, J.M.: Complementarity modeling of hybrid systems. IEEE Trans. Automat. Contr. AC-43, 483–490 (1998)Google Scholar
  49. 49.
    van der Schaft, A.J., Schumacher, J.M.: An Introduction to Hybrid Dynamical Systems. Springer, London, 2000Google Scholar
  50. 50.
    Schrijver, A.: Theory of Linear and Integer Programming. Wiley, Chichester, 1986Google Scholar
  51. 51.
    Schumacher, J.M.: State representations of linear systems with output constraints. Math. Control, Signals Syst. 3, 61–80 (1990)Google Scholar
  52. 52.
    Sontag, E.D.: Mathematical Control Theory: Deterministic Finite Dimensional Systems. Springer Verlag, New York, 1990Google Scholar
  53. 53.
    Stewart, D.E.: A high accuracy method for solving ODEs with discontinuous right-hand side. Numer. Math. 58, 299–328 (1990)MathSciNetMATHGoogle Scholar
  54. 54.
    Stewart, D.E.: Convergence of a time-stepping scheme for rigid body dynamics and resolution of Painlevé’s problem. Arch. Ration. Mech. Anal. 145, 215–260 (1998)CrossRefMathSciNetMATHGoogle Scholar
  55. 55.
    Trentelman, H.L., Stoorvogel, A.A., Hautus, M.L.J.: Control Theory for Linear Systems. Springer, London, 2001Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  1. 1.Department of Econometrics and Operations ResearchTilburg UniversityLE TilburgThe Netherlands

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