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A simplicial algorithm for the nonlinear complementarity problem

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Abstract

A triangulation of the nonnegative orthant and a special labeling of the vertices lead to a combinatorial procedure for seeking solutions or approximate solutions to the nonlinear complementarity problem under coercive-like assumptions on the problem functions. Derivatives are not required. Convergence is proved, computational considerations are discussed, and some preliminary applications to convex programming and saddle point computation, along with numerical results, are presented.

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References

  1. A.R. Colville, “A comparative study of nonlinear programming codes”, IBM New York Scientific Center Rept. No. 320-2949 (1968).

  2. R.W. Cottle, “Nonlinear programs with positively bounded Jacobians”,SIAM Journal of Applied Mathematics 14 (1966) 147–158.

    Google Scholar 

  3. R.W. Cottle and G.B. Dantzig, “Complementary pivot theory of mathematical programming”,Linear Algebra and its Applications 1 (1968) 103–125.

    Google Scholar 

  4. B.C. Eaves, “Computing Kakutani fixed points”,SIAM Journal of Applied Mathematics 21 (1971) 236–244.

    Google Scholar 

  5. B.C. Eaves and R. Saigel, “Homotopies for computation of fixed points on unbounded regions”,Mathematical Programming 3 (1972) 225–237.

    Google Scholar 

  6. C.B. Garcia, C.E. Lemke and H. Luethi, “Simplicial approximation of an equilibrium point for non-cooperativeN-person games”, in:Mathematical programming Eds. T.C. Hu and S. Robinson (Academic Press, New York, 1973).

    Google Scholar 

  7. F.J. Gould and J.W. Tolle, “A unified approach to complementarity in optimization”,Discrete Mathematics 7 (1974) 225–271.

    Google Scholar 

  8. S. Karamardian, “The complementarity problem”,Mathematical Programming 2 (1972) 107–129.

    Google Scholar 

  9. H.W. Kuhn, “Simplicial approximation of fixed points”,Proceedings of the National Academy of Sciences 61 (1968) 1238–1242.

    Google Scholar 

  10. H.W. Kuhn, “Some combinatorial lemmas in topology”,IBM Journal of Research and Development 4 (1960) 518–524.

    Google Scholar 

  11. C.E. Lemke, “On Complementary pivot theory”, in:Mathematics of the decision sciences Eds. G.B. Dantzig and A.F. Veinott, Jr. (Am. Math. Soc., Providence, R.I., 1968) pp. 95–115.

    Google Scholar 

  12. O.H. Merrill, “Applications and extensions of an algorithm that computes fixed points of certain non-empty convex upper semi-continuous point to set mappings”, Dept. of Industrial Engineering, University of Michigan, Ann Arbor, Mich., Tech. Rept. No. 71-7 (September 1971).

    Google Scholar 

  13. J.B. Rosen and S. Suzuki, “Construction of nonlinear programming test problems”,Communications of the Association for Computing Machinery 8 p. 113.

  14. H. Scarf, “The approximation of fixed points of a continuous mapping”,SIAM Journal on Applied Mathematics 15 (1967) 1328–1343.

    Google Scholar 

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The work of this author was sponsored in part by National Science Foundation grant GJ-1154X to the National Bureau of Economic Research, Inc., while the author was a Research Associate at the National Bureau's Computer Research Center for Economics and Management Science of Cambridge, Mass.

The work of this author was sponsored in part by the Office of Naval Research, Contract No. N000-14-67-A-0321-0003 (NR-047-085).

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Fisher, M.L., Gould, F.J. A simplicial algorithm for the nonlinear complementarity problem. Mathematical Programming 6, 281–300 (1974). https://doi.org/10.1007/BF01580246

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  • DOI: https://doi.org/10.1007/BF01580246

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