Skip to main content
SpringerLink
Log in
Book cover

Point-to-Set Maps and Mathematical Programming pp 142–157Cite as

The fixed point approach to nonlinear programming

  • Chapter
  • First Online:

Part of the book series: Mathematical Programming Studies ((MATHPROGRAMM,volume 10))

Abstract

In this paper we consider the application of the recent algorithms that compute fixed points in unbounded regions to the nonlinear programming problem. It is shown that these algorithms solve the inequality constrained problem with functions that are not necessarily differentiable. The application to convex and piecewise linear problems is also discussed.

This research was partially supported by Grant no. mes 77-03472 from the National Science Foundation.

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

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. F.H. Clarke, “Generalized gradients and applications”, Transactions of the American Mathematical Society 205 (1975) 247–262.

    Article  MATH  MathSciNet  Google Scholar 

  2. F.H. Clarke, “A new approach to Lagrange multipliers”, Mathematics of Operations Research 1 (1976) 165–174.

    Article  MATH  MathSciNet  Google Scholar 

  3. B.C. Eaves, “Nonlinear programming via kakutani fixed points” Working Paper No. 294, Center for Research in Management Science, University of California, Berkeley, 1970.

    Google Scholar 

  4. B.C. Eaves and R. Saigal, “Homotopies for computation of fixed points on unbounded regions,” Mathematical Programming 8 (1972) 134–145.

    MathSciNet  Google Scholar 

  5. T. Hansen, “A fixed point algorithm for approximating the optimal solution of a concave programming problem,” Cowles Foundation Discussion Paper No. 277, Yale University, New Haven, Connecticut, 1969.

    Google Scholar 

  6. T. Hansen and H. Scarf, “On the applications of a recent combinational algorithm,” Cowles Foundation Discussion Paper No. 272, Yale University, New Haven, Connecticut, 1969.

    Google Scholar 

  7. O.H. Merrill, “Applications and extensions of an algorithm that computes fixed points of certain upper-semi continuous point to set mapping,” Ph.D. Dissertation, University of Michigan, Ann Arbor, 1972.

    Google Scholar 

  8. R. Mifflin, “An algorithm for constrained optimization with semismooth functions”, Mathematics of Operations Research 2 (1977) 191–207.

    Article  MATH  MathSciNet  Google Scholar 

  9. A. Netravali and R. Saigal, “Optimum quantizer design using a fixed point algorithm”, The Bell System Technical Journal 55 (1976) 1423–1435.

    MathSciNet  Google Scholar 

  10. R.T. Rockafellar, Convex analysis (Princeton University Press, Princeton, NJ, 1970).

    MATH  Google Scholar 

  11. D.G. Saari and R. Saigal, in preparation.

    Google Scholar 

  12. R. Saigal, “Investigations into the efficiency of fixed point algorithms,” in: S. Karamardian, ed., Fixed points—algorithms and applications (Academic Press, New York, 1977).

    Google Scholar 

  13. R. Saigal, “Fixed point computing methods,” Encyclopedia of Computer Science and Technology, Vol. 8 (Marcel Dekker, New York, 1977).

    Google Scholar 

  14. R. Saigal, “On paths generated by fixed point algorithms”, Mathematics of Operations Research 1 (1976) 359–380.

    Article  MATH  MathSciNet  Google Scholar 

  15. R. Saigal, “On the convergence rate of algorithms for solving equations that are based on methods of complementary pivoting”, Mathematics of Operations Research 2 (1977) 108–124.

    Article  MATH  MathSciNet  Google Scholar 

  16. H. Scarf, “The approximation of fixed points of a continuous mapping, SIAM Journal of Applied Mathematics 16 (1967) 1328–1343.

    Article  MathSciNet  Google Scholar 

  17. M.J. Todd, “On triangulations for computing fixed points”, Mathematical Programming 10 (1976) 322–346.

    Article  MATH  MathSciNet  Google Scholar 

  18. M.J. Todd, The computation of fixed points and applications (Springer-Verlag, Berlin-Heidelburg, 1976).

    MATH  Google Scholar 

  19. M.J. Todd, “New fixed point algorithms for economic equilibria and constrained optimization,” Tech. Report No. 362, School of Operations Research, Cornel University, Ithaca, New York. 1977.

    Google Scholar 

  20. P. Wolfe, “A method of conjugate subgradients for minimizing nondifferentiable functions,” in: M.L. Balinski and P. Wolfe, eds., Nondifferentiable optimization, Mathematical Programming Study 3 (North-Holland, Amsterdam, 1975) 145–173.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Northwestern University, 60201, Evanston, IL, USA

    R. Saigal

Authors
  1. R. Saigal
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

P. Huard

Rights and permissions

Reprints and permissions

Copyright information

© 1979 The Mathematical Programming Society

About this chapter

Cite this chapter

Saigal, R. (1979). The fixed point approach to nonlinear programming. In: Huard, P. (eds) Point-to-Set Maps and Mathematical Programming. Mathematical Programming Studies, vol 10. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0120851

Download citation

  • DOI: https://doi.org/10.1007/BFb0120851

  • Received:

  • Revised:

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-00797-2

  • Online ISBN: 978-3-642-00798-9

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation