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Simplicial approximation of solutions to the nonlinear complementarity problem with lower and upper bounds

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Abstract

Ideas of a simplicial variable dimension restart algorithm to approximate zero points onR n developed by the authors and of a linear complementarity problem pivoting algorithm are combined to an algorithm for solving the nonlinear complementarity problem with lower and upper bounds. The algorithm can be considered as a modification of the2n-ray zero point finding algorithm onR n. It appears that for the new algorithm the number of linear programming pivot steps is typically less than for the2n-ray algorithm applied to an equivalent zero point problem. This is caused by the fact that the algorithm utilizes the complementarity conditions on the variables.

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References

  1. E.L. Allgower and K. Georg, “Simplicial and continuation methods for approximating fixed points and solutions to systems of equations,”SIAM Review 22 (1980) 28–85.

    Article  MATH  MathSciNet  Google Scholar 

  2. S.N. Chow, J. Mallet-Paret and J.A. Yorke, “Finding zeroes of maps: homotopy methods that are constructive with probability one,”Mathematics of Computation 32 (1978) 887–899.

    Article  MATH  MathSciNet  Google Scholar 

  3. B.C. Eaves, “On the basic theory of complementarity,”Mathematical Programming 1 (1971) 68–75.

    Article  MATH  MathSciNet  Google Scholar 

  4. M.L. Fisher and F.J. Gould, “A simplicial algorithm for the nonlinear complementarity problem,”Mathematical Programming 6 (1974) 281–300.

    Article  MATH  MathSciNet  Google Scholar 

  5. C.B. Garcia, “The complementarity problem and its applications,” Ph.D. Thesis, Rensselaer Polytechnic Institute, Troy, NY (1973).

    Google Scholar 

  6. G.J. Habetler and K.M. Kostreva, “On a direct algorithm for nonlinear complementarity problems,”SIAM Journal of Control and Optimization 16 (1978) 504–511.

    Article  MATH  MathSciNet  Google Scholar 

  7. M. Kojima, “Computational methods for solving the nonlinear complementarity problem”, Keio Engineering Reports 27, Keio University, Yokohama, Japan (1974).

    Google Scholar 

  8. M. Kojima and R. Saigal, “On the number of solutions to a class of complementarity problems,”Mathematical Programming 21 (1981) 190–203.

    Article  MATH  MathSciNet  Google Scholar 

  9. M. Kojima and Y. Yamamoto, “Variable dimension algorithms: Basic theory, interpretations and extensions of some existing methods,”Mathematical Programming 24 (1982) 177–215.

    Article  MATH  MathSciNet  Google Scholar 

  10. M. Kojima and Y. Yamamoto, “A unified approach to the implementation of several restart fixed point algorithms and a new variable dimension algorithm,”Mathematical Programming 28 (1984) 288–328.

    MATH  MathSciNet  Google Scholar 

  11. G. van der Laan and A.J.J. Talman, “A restart algorithm for computing fixed points without an extra dimension,”Mathematical Programming 17 (1979) 74–84.

    Article  MATH  MathSciNet  Google Scholar 

  12. G. van der Laan and A.J.J. Talman, “A class of simplicial restart fixed point algorithms without an extra dimension,”Mathematical Programming 20 (1981) 33–48.

    Article  MATH  MathSciNet  Google Scholar 

  13. G. van der Laan and A.J.J. Talman, “Simplicial algorithms for finding stationary points, a unifying description,”Journal of Optimization Theory and Applications 50 (1986) 165–182.

    Article  MATH  MathSciNet  Google Scholar 

  14. H.J. Lüthi, “A simplicial approximation of a solution for the nonlinear complementarity problem,”Mathematical Programming 9 (1975) 278–293.

    Article  MATH  MathSciNet  Google Scholar 

  15. O.L. Mangasarian, “Equivalence of the complementarity problem to a system of nonlinear equations,”SIAM Journal of Applied Mathematics 31 (1976) 89–92.

    Article  MathSciNet  MATH  Google Scholar 

  16. O.H. Merrill, “Applications and extensionof an algorithm that computes fixed points of certain upper semi-continuous point-to-set mappings” Ph.D. Thesis, University of Michigan, Ann Arbor, Mich. (1972).

    Google Scholar 

  17. J.J. Moré, “Coercivity conditions in nonlinear complementarity algorithms”,SIAM Review 16 (1974) 1–15.

    Article  MATH  MathSciNet  Google Scholar 

  18. R. Saigal, “A homotopy for solving large, sparse and structured fixed point problems”,Mathematics of Operations Research 8 (1983) 557–578.

    Article  MATH  MathSciNet  Google Scholar 

  19. A.J.J. Talman, “Variable dimension fixed point algorithms and triangulations,” Mathematical Centre Tracts, 128 (Mathematisch Centrum, Amsterdam, 1980).

    MATH  Google Scholar 

  20. A.J.J. Talman and L. Van der Heyden, “Algorithms for the linear complementarity problem which allow an arbitrary starting point,” in: B.C. Eaves et al., eds.Homotopy Methods and Global Convergence (Plenum Press, New York, 1983) pp. 267–286.

    Google Scholar 

  21. M.J. Todd, “Improving the convergence of fixed point algorithms”,Mathematical Programming Study 7 (1978) 151–169.

    MATH  MathSciNet  Google Scholar 

  22. M.J. Todd, “Global and local convergence and monotonicity results for a recent variable dimension simplicial algorithm,” in W. Forster, ed.,Numerical Solution of Highly Nonlinear Problems (North-Holland, Amsterdam, 1980) pp. 43–69.

    Google Scholar 

  23. L.T. Watson, “Solving the nonlinear complementarity problem by a homotopy method,”SIAM Journal of Control and Optimization 17 (1979) 36–46.

    Article  MATH  Google Scholar 

  24. A.H. Wright, “The octahedral algorithm, a new simplicial fixed point algorithm”,Mathematical Programming 21 (1981) 47–9.

    Article  MATH  MathSciNet  Google Scholar 

  25. Y. Yamamoto, “A new variable dimension algorithm for the fixed point problem”,Mathematical Programming 25 (1983) 329–342.

    MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Economics and Econometrics, Free University, Amsterdam, The Netherlands

    G. van der Laan

  2. Department of Econometrics, Tilburg University, Tilburg, The Netherlands

    A. J. J. Talman

Authors
  1. G. van der Laan
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  2. A. J. J. Talman
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Additional information

This work is part of the VF-program “Equilibrium and Disequilibrium in Demand and Supply,” which has been approved by the Netherlands Ministry of Education and Sciences.

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van der Laan, G., Talman, A.J.J. Simplicial approximation of solutions to the nonlinear complementarity problem with lower and upper bounds. Mathematical Programming 38, 1–15 (1987). https://doi.org/10.1007/BF02591848

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  • DOI: https://doi.org/10.1007/BF02591848

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