Skip to main content
SpringerLink
Log in

Homotopies for computation of fixed points on unbounded regions

  • Published:
Mathematical Programming Submit manuscript

Abstract

Givenf: R n → R n* with some conditions, our aim is to compute a fixed pointx ∈ f(x) off; hereR n isn-dimensional Euclidean space andR n* is the collection of nonempty subsets ofR n. A typical application of the algorithm can be motivated as follows: Beginning with the constant mapf 0:R n → {0} ⊂R n and its fixed pointx 0 = 0, we deformf t ast → ∞ tof ∞ ∈ f and follow the pathx t of fixed points off t . Cluster points of thex t 's ast → ∞ are fixed points off.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. C. Berge,Topological spaces (MacMillan Co., New York, 1963).

    Google Scholar 

  2. B.C. Eaves, “Computing Kakutani fixed points,”SIAM Journal of Applied Mathematics, 21 (2) (1971) 236–244.

    Google Scholar 

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

    Google Scholar 

  4. B.C. Eaves, “Homotopies for computation of fixed points,”Mathematical Programming 3 (1972) 1–22.

    Google Scholar 

  5. J. Freidenfields, “A fixed point algorithm and almost-complementary sets,” Technical Report No. 71-3, Department of Operations Research, Stanford University, April 1971. In addition, “Fixed point algorithms and almost-complementary sets,” Ph.D. dissertation, Department of Operations Research, Stanford University, August 1971.

  6. T. Hansen, “On the approximation of a competitive equilibrium,” Ph.D. dissertation, Yale University, 1968.

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

    Google Scholar 

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

    Google Scholar 

  9. C.E. Lemke, “Bimatrix equilibrium points and mathematical programming,”Management Science 11 (7) (1965) 681–689.

    Google Scholar 

  10. 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,” Department of Industrial Engineering, University of Michigan, Technical Report No. 71-7, September 1971.

  11. H. Scarf, “The approximation of fixed points of a continuous mapping,”SIAM Journal of Applied Mathematics 15 (5) (1967).

  12. E.H. Spanier,Algebraic topology (McGraw-Hill, New York, 1966).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Stanford University, Stanford, California, USA

    B. Curtis Eaves

  2. University of California, Berkeley, California, USA

    Romesh Saigal

Authors
  1. B. Curtis Eaves
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Romesh Saigal
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

This research was supported in part by Army Research Office-Durham Contract DAHC-04-71-C-0041 and by National Science Foundation Grant GK-5695.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Eaves, B.C., Saigal, R. Homotopies for computation of fixed points on unbounded regions. Mathematical Programming 3, 225–237 (1972). https://doi.org/10.1007/BF01584991

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01584991

Keywords

Search

Navigation