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On the solution of large, structured linear complementarity problems: The tridiagonal case

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Abstract

This paper delineates the underlying theory of an efficient method for solving a class of specially-structured linear complementarity problems of potentially very large size. Problems of the type considered here arise in the process of making discrete approximations to differential equations in the presence of special side conditions. This problem source is exemplified by the free boundary problem for (infinite) journal bearings. Some of the authors' computational experience with the method is presented.

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References

  1. J. C. Bierlein, The Journal Bearing,Scientific American 233 (1975), 50–64.

    Google Scholar 

  2. A. Cameron andW. L. Wood, The Full Journal Bearing,Inst. Mech. Engrs. J. Proc. 161 (1949), 59–64.

    Google Scholar 

  3. R. Chandrasekaran, A Special Case of the Complementary Pivot Problem,Opsearch 7 (1970), 263–268.

    Google Scholar 

  4. D. G. Christopherson, A New Mathematical Method for the Solution of Film Lubrication Problems,Inst. Mech. Engrs. J. Proc. 146 (1941), 126–135.

    Google Scholar 

  5. R. W. Cottle, Nonlinear Programs with Positively Bounded Jacobians,Doctoral Dissertation, University of California, Berkeley, 1964.

    Google Scholar 

  6. R. W. Cottle, Nonlinear Programs with Positively Bounded Jacobians,J. SIAM Appl. Math. 14 (1966), 147–158.

    Article  Google Scholar 

  7. R. W. Cottle, Principal Pivoting Method of Quadratic Programming,Mathematics of the Decision Sciences, Part I (G. B. Dantzig and A. F. Veinott, Jr., eds.), American Mathematical Society, Providence, R. I., 1968.

    Google Scholar 

  8. R. W. Cottle, Monotone Solutions of the Parametric Linear Complementarity Problem,Math. Prog. 3 (1972), 210–224.

    Google Scholar 

  9. R. W. Cottle andG. B. Dantzig, Complementary Pivot Theory of Mathematical Programming,Linear Algebra and Appl. 1 (1968), 103–125.

    Article  Google Scholar 

  10. C. W. Cryer, The Method of Christopherson for Solving Free Boundary Problems for Infinite Journal Bearings by Means of Finite Differences,Math. of Computation 25 (1971), 435–443.

    Google Scholar 

  11. C. W. Cryer, The Solution of a Quadratic Programming Problem Using Systematic Overrelaxation,J. SIAM Control 9 (1971), 385–392.

    Article  Google Scholar 

  12. G. B. Dantzig,Linear Programming and Extensions, Princeton University Press, Princeton, 1963.

    Google Scholar 

  13. G. B. Dantzig andR. W. Cottle, Positive Semi-Definite Programming,Nonlinear Programming (J. Abadie, ed.), North Holland Publishing Company, Amsterdam, 1967.

    Google Scholar 

  14. M. A. Diamond, The Solution of a Quadratic Programming Problem Using Fast Methods to Solve Systems of Linear Equations,Internat. J. Systems Sci. 5 (1974), 131–136.

    Google Scholar 

  15. M. Fiedler andV. Ptak, On Matrices with Nonpositive Off-Diagonal Elements and Positive Principal Minors,Czech. J. Math. 12 (1962), 382–400.

    Google Scholar 

  16. A. A. Gnanadoss andM. R. Osborne, The Numerical Solution of Reynolds' Equation for a Journal Bearing,Quart. J. Mech. Appl. Math. 17 (1964), 241–246.

    Google Scholar 

  17. M. D. Hersey,Theory and Research in Lubrication, J. Wiley and Sons, New York, 1966.

    Google Scholar 

  18. E. Isaacson andH. B. Keller,Analysis of Numerical Methods, J. Wiley and Sons, New York, 1966.

    Google Scholar 

  19. E. L. Keller, Quadratic Optimization and Linear Complementarity,Doctoral Dissertation, University of Michigan, 1969. See alsoMath. Programming 5 (1973), 331–337.

    Google Scholar 

  20. C. E. Lemke, On Complementary Pivot Theory,Mathematics of the Decision Sciences, Part I (G. B. Dantzig and A. F. Veinott, Jr., eds.) American Mathematical Society, Providence, R. I., 1968.

    Google Scholar 

  21. C. E. Lemke, Recent Results on Complementarity Problems,Nonlinear Programming (J. B. Rosen, O. L. Mangasarian, and K. Ritter, eds.), Academic Press, New York, 1970.

    Google Scholar 

  22. C. E. Lemke andJ. T. Howson, Equilibrium Points of Bimatrix Games,J. Soc. Indust. Appl. Math. 12 (1964), 413–442.

    Article  Google Scholar 

  23. O. Pinkus andB. Sternlicht,Theory of Hydrodynamic Lubrication, McGraw-Hill, New York, 1961.

    Google Scholar 

  24. A. A. Raimondi andJ. Boyd, A Solution for the Finite Journal Bearing and Its Application to Analysis and Design: III,Trans. Amer. Soc. Lubrication Engrs. 1 (1958), 194–209.

    Google Scholar 

  25. R. Saigal, A Note on a Special Linear Complementarity Problem,Opsearch 7 (1970), 175–183.

    Google Scholar 

  26. R. Saigal, Lemke's Algorithm and a Special Linear Complementarity Problem,Opsearch 8 (1971), 201–208.

    Google Scholar 

  27. H. Samelson, R. M. Thrall, andO. Wesler, A Partition Theorem for Euclideann-Space,Proc. Amer. Math. Soc. 9 (1958), 805–807.

    Google Scholar 

  28. D. M. Smith,Journal Bearing in Turbomachinery, Chapman and Hall, Ltd., London, 1969.

    Google Scholar 

  29. A. W. Tucker, A Combinatorial Equivalence of Matrices,Proceedings of Symposia in Applied Mathematics, Vol. 10 (R. Bellman and M. Hall, eds.), American Mathematical Society, Providence, R. I., 1960.

    Google Scholar 

  30. A. W. Tucker, Principal Pivotal Transforms of Square Matrices,SIAM Review 5 (1963), p. 305.

    Google Scholar 

  31. R. S. Varga,Matrix Iterative Analysis, Prentice Hall, Englewood Cliffs, N.J., 1962.

    Google Scholar 

  32. D. F. Wilcock andE. R. Booser,Bearing Design and Application, McGraw-Hill, New York, 1957.

    Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Operations Research, Stanford University, 94305, Stanford, California, USA

    Richard W. Cottle

  2. Department of Mathematical Sciences, Rensselaer Polytechnic Institute, 12181, Troy, New York, USA

    Richard S. Sacher

Authors
  1. Richard W. Cottle
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  2. Richard S. Sacher
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Additional information

Communicated by G. Golub

Research and reproduction of this report was partially supported by the Office of Naval Research under contract N-00014-67-A-0112-0011; U.S. Atomic Energy Commission Contract AT(04-3)-326 PA # 18.

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Cottle, R.W., Sacher, R.S. On the solution of large, structured linear complementarity problems: The tridiagonal case. Appl Math Optim 3, 321–340 (1976). https://doi.org/10.1007/BF01448184

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  • DOI: https://doi.org/10.1007/BF01448184

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