Mathematical Programming

, Volume 3, Issue 1, pp 1–22 | Cite as

Homotopies for computation of fixed points

  • B. Curtis Eaves


Given a point to set mapf on a simplex with certain conditions, an algorithm for computing fixed points is described. The algorithm operates by following the fixed point as an initially affine function is deformed towardsf.


Mathematical Method Affine Function 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    F.E. Browder, “On continuity of fixed points under deformations of continuous mappings,”Summa Brasilia Mathematica 4, 5 (1960) 183, 191.Google Scholar
  2. [2]
    G.B. Dantzig,Linear programming and extensions (Princeton University Press, Princeton, N.J., 1963).Google Scholar
  3. [3]
    B.C. Eaves, “An odd theorem,”Proceedings of the American Mathematical Society 26, 3 (1970) 509–513.Google Scholar
  4. [4]
    B.C. Eaves, “Computing Kakutani fixed points,”SIAM Journal of Applied Mathematics 21, 2 (1971) 236–244.Google Scholar
  5. [5]
    B.C. Eaves, “On the basic theorem of complementarity,”Mathematical Programming 1 (1971) 68–75.Google Scholar
  6. [6]
    S. Eilenberg and N. Steenrod,Foundations of algebraic topology (Princeton University Press, Princeton, N.J., 1952).Google Scholar
  7. [7]
    T. Hansen, “On the approximation of a competitive equilibrium,” dissertation, Yale University, New Haven, 1968.Google Scholar
  8. [8]
    T. Hansen and H. Scarf, “On the applications of a recent combinatorial algorithm,” (Cowles Foundation Discussion Paper No. 272, April 1969).Google Scholar
  9. [9]
    H.W. Kuhn, “Some combinatorial lemmas in topology,”IBM J. Research and Development 4 (1960) 518–524.Google Scholar
  10. [10]
    H.W. Kuhn, “Simplicial approximation of fixed points,”Proceedings of the National Academy of Sciences 61 (1968) 1238–1242.Google Scholar
  11. [11]
    C.E. Lemke, “Bimatrix equilibrium points and mathematical programming,”Management Science 11, 7 (1965) 681–689.Google Scholar
  12. [12]
    C.E. Lemke and J.T. Howson, Jr., “Equilibrium points of bimatrix games,”Journal of the Society of Industrial Applied Mathematics 12, 2 (1964) 413–423.Google Scholar
  13. [13]
    H. Scarf, “The approximation of fixed points of a continuous mapping,”SIAM Journal of Applied Mathematics 15, 5 (1967) 1328–1343.Google Scholar
  14. [14]
    E.H. Spanier,Algebraic topology (McGraw-Hill, New York, 1966).Google Scholar

Copyright information

© The Mathematical Programming Society 1972

Authors and Affiliations

  • B. Curtis Eaves
    • 1
  1. 1.Stanford UniversityStanfordUSA

Personalised recommendations