Parametrization and Reduction to Nonlinear Equations

  • G. Isac
  • V. A. Bulavsky
  • V. V. Kalashnikov
Chapter
Part of the Nonconvex Optimization and Its Applications book series (NOIA, volume 63)

Abstract

This chapter is dedicated to two different types of methods solving nonlinear complementarity problems. The first one is a method of approximate solution of nonlinear complementarity problem with parameters: Given a continuous mapping f : R n × R m → R n , and a fixed vector of parameters u = (u 1, ..., u m ) T , find a x ∈ R n such that
$$ x \ge 0,\quad f(x,u) \ge 0,\quad and\quad {x^T}f(x,u) = 0 $$
(11.1)
.

Keywords

Complementarity Problem Regularization Method Linear Complementarity Problem Merit Function Nonlinear Complementarity Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Billups SC. “Algorithms for complementarity problems and generalized equations”. PhD Thesis, Computer Science Department, University of Wisconsin, Madison, WI, 1995.Google Scholar
  2. Billups SC and Ferris MC. QPCOMP: A quadratic programming based solver for mixed complementarity problems. Math. Programming. 1997; 76: 533–562.MathSciNetMATHGoogle Scholar
  3. Chen B and Harker PT. 1. A noninterior continuation method for quadratic and linear programming. SIAM J. Optim. 1993; 3: 503–515.MathSciNetMATHCrossRefGoogle Scholar
  4. Chen B and Harker PT. 2. Smooth approximations to nonlinear complementarity problems. SIAM J. Optim. 1997; 7: 402–420.MathSciNetGoogle Scholar
  5. Chen B, Chen X and Kanzow C. A penalized Fisher-Burmeister NCP-functioni Theoretical investigation and numerical results. Math. Programming, 2000; 88: 211–216.MathSciNetMATHCrossRefGoogle Scholar
  6. Cottle, R.W., Pang, J.-S. and Stone, R.E. The Linear Complementarity Problem. Boston: Academic Press, 1992.MATHGoogle Scholar
  7. De Luca T, Facchinei F, Kanzow C. A semismooth equation approach to the solution of nonlinear complementarity problems. Math. Programming, 1996; 75: 407–439.MathSciNetMATHCrossRefGoogle Scholar
  8. Dontchev, A.L. and Zolezzi, T. Well-Posed Optimization Problems. Lecture Notes in Mathematics No. 1543, Springer-Verlag, 1993.MATHGoogle Scholar
  9. Ebiefung A A. New perturbation results for solving the linear complementarity problem with P 0 -matrices. Appl. Math. Letters, 1998; 11: 37–39.MathSciNetCrossRefGoogle Scholar
  10. Facchinei F. Structural and stability properties of P 0 nonlinear complementarity problems. Math. Oper. Res. 1998; 23: 735–745.MathSciNetMATHCrossRefGoogle Scholar
  11. Facchinei F and Kanzow C. 1. A nonsmooth inexact Newton method for the solution of large-scale nonlinear complementarity problems. Math. Programming, 1997; 76: 493–512.MathSciNetMATHGoogle Scholar
  12. Facchinei F and Kanzow C. 2. Beyond monotonicity in regularization methods for nonlinear complementarity problems. SIAM J. Control Optim. 1999; 37: 1150–1161.MathSciNetMATHCrossRefGoogle Scholar
  13. Facchinei F, Soares J. A new merit function for nonlinear complementarity problems and a related algorithm. SIAM J. Optim. 1997; 7: 225–247.MathSciNetMATHCrossRefGoogle Scholar
  14. Ferris, M.C, Kanzow C. ‘Complementarity and related problems.’ — In: Handbook on Applied Optimization, P.M. Pardalos and M.G.C. Resende, eds. Oxford: Oxford University Press (to appear).Google Scholar
  15. Fischer A. A special Newton-type optimization method. Optimization, 1992; 22: 269–284.CrossRefGoogle Scholar
  16. Geiger C, Kanzow C. On the resolution of monotone complementarity problems. Comput. Optim. Appl. 1996; 5: 155–173.MathSciNetMATHCrossRefGoogle Scholar
  17. Gowda MS and Tawhid MA. “Existence and limiting behavior of trajectories associated with P 0-equations.” Research Report 97–15, Department of Mathematics and Statistics, University of Maryland Baltimore County, Baltimore, Maryland 21250, USA, November 1997.Google Scholar
  18. Harker PT and Pang JS. Finite-dimensional variational inequality and nonlinear complementarity problems: a survey of theory, algorithms and applications. Math. Programming, 1990; 48: 161–220.MathSciNetMATHCrossRefGoogle Scholar
  19. Kalashnikov VV. Existence theorem for nonlinear complementarity problem. Optimization (Novosibirsk, Institute of Matehmatics). 1989; 45(62): 26–33. (in Russian). MathSciNetGoogle Scholar
  20. Kalashnikov VV and Kalashnikova NI. 1. Convergence of Newton method for solving nonlinear complementarity problems. Optimization (Novosibirsk, Institute of Mathematics), 1991; 49(65): 69–79 (in Russian). MathSciNetGoogle Scholar
  21. Kalashnikov VV and Kalashnikova NI. 2. An iteration method to solve nonlinear complementarity problems. Optimization (Novosibirsk, Institute of Mathematics), 1993; 52(69): 42–54 (in Russian). MathSciNetGoogle Scholar
  22. Kanzow C. Global convergence properties of some iterative methods for linear complementarity problems. SIAM J. Optim. 1996; 6: 326–341.MathSciNetMATHCrossRefGoogle Scholar
  23. Kanzow C and Kleinmichel H. A new class of semismooth Newton-type methods for nonlinear complementarity problems. Comput. Optim. Appl. 1998; 11: 227–251.MathSciNetMATHCrossRefGoogle Scholar
  24. Kanzow C and Pieper H. Jacobian smoothing methods for nonlinear complementarity problems. SIAM J. Optim. 1999; 9: 342–372.MathSciNetMATHCrossRefGoogle Scholar
  25. Kanzow C and Zupke M. ‘Inexact trust region methods for nonlinear complementarity problems.’ — In: Reformulation: Nonsmooth, Piecewise Smooth, Semismooth and Smoothing Methods, M. Fukushima and L. Qi, eds. Dordrecht: Kluwer Academic Press, 1999. — P. 211–233.Google Scholar
  26. Kyparisis J. Uniqueness and differentiability of parametric nonlinear complementarity problems. Math. Programming, 1986; 36: 105–113.MathSciNetMATHCrossRefGoogle Scholar
  27. McLinden, L. ‘The Complementarity Problem for Maximal Monotone Multifiinctions’.— In: R.W. Cottle, F. Giannessi, and J.L. Lions, eds. Variational Inequalities and Complementarity Problems. New York: Academic Press, 1980.- P. 251–270.Google Scholar
  28. Moré J J. 1. Coercivity conditions in nonlinear complementarity problems. SIAM Review, 1974; 16: 1–16.MathSciNetMATHCrossRefGoogle Scholar
  29. Moré J J. 2. Classes of functions and feasibility conditions in nonlinear complementarity problems. Math. Programming, 1974; 6: 327–338.MathSciNetMATHCrossRefGoogle Scholar
  30. More JJ and Rheinboldt WC. On P- and S-functions and related classes of n-dimensional nonlinear mappings. Lin. Alg. Appl. 1973; 6: 45–68.MathSciNetMATHCrossRefGoogle Scholar
  31. Ortega, J., Rheinboldt, W.C. Iterative Solution of Nonlinear Equations in Several Variables. New York: Academic Press, 1970.MATHGoogle Scholar
  32. Palais, R.S. and Terng, C.-L. Critical Point Theory and Submanifold Geometry. Lecture Notes in Mathematics, No. 1353. Berlin: Springer-Verlag, 1988.MATHGoogle Scholar
  33. Polyak, B.T. Introduction to Optimization. New York: Optimization Software Inc., 1987.Google Scholar
  34. Qi HD. “A regularized smoothing Newton method for box constrained variational inequality problems with PQ functions.” Research Report, Chinese Academy of Sciences, Institute of Computational Mathematics and Scientific/Engineering Computing, Beijing, China, July 1997.Google Scholar
  35. Ravindran G and Gowda MS. “Regularization of P 0-functions in box variational inequality problems.” Research Report 97–07, Department of Mathematics and Statistics, University of Maryland Baltimore County, Baltimore, Maryland 21250, USA, August (revised October), 1997.Google Scholar
  36. Subramanyan PK. A note on least two norm solutions of monotone complementarity problems. Appl. Math. Letters, 1988; 1: 395–397.CrossRefGoogle Scholar
  37. Sun D. “A regularization Newton method for solving nonlinear complementarity problems”. Research Report, School of Mathematics, The University of New South Wales, Sydney 2052, Australia, July 1997.Google Scholar
  38. Sznajder R and Gowda MS. “On the limiting behavior of the trajectory of regularized solutions of a P 0-complementarity problem”. Research Report 97–08, Department of Mathematics and Statistics, University of Maryland Baltimore County, Baltimore, Maryland 21250, USA, October, 1997.Google Scholar
  39. Tseng P. An infeasible path-following method for monotone complementarity problems. SIAM J. Optim. 1997; 7: 386–402.MathSciNetMATHCrossRefGoogle Scholar
  40. Venkateswaran V. An algorithm for the linear complementarity problem with a P 0-matrix. SIAM J. Matrix Anal. Appl. 1993; 14: 967–977.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2002

Authors and Affiliations

  • G. Isac
    • 1
  • V. A. Bulavsky
    • 2
  • V. V. Kalashnikov
    • 2
  1. 1.Department of Mathematics and Computer ScienceRoyal Military College of CanadaKingstonCanada
  2. 2.Central Economics Institute (CEMI) of Russian Academy of SciencesMoscowRussia

Personalised recommendations