Skip to main content
SpringerLink
Log in

Asymptotic properties of sequences iteratively generated by point-to-set maps

  • Chapter
  • First Online:
Book cover Point-to-Set Maps and Mathematical Programming

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

Abstract

In this paper, we examine the relationships between the fixed point set of a point-to-set map A(·), and the asymptotic properties of the sequences which may be iteratively generated by using the map A(·). Let L be the set of all limit points, and Q be the set of all cluster points of all sequences which may be iteratively generated by A(·). The consequences of various assumptions on the map A(·) and the sequences generated by A(·) on lower-bounds and upper-bounds for L and Q are discussed.

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

Access this chapter

Log in via an institution

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. C. Berge, Espaces topologiques—functions multivoques (Dunod, Paris, 1966).

    Google Scholar 

  2. G. Debreu, Theory of Value, Cowles Foundations, Monograph 17, (Wiley, New York, 1959).

    MATH  Google Scholar 

  3. B.C. Eaves and W.I. Zangwill, “Generalized cutting plane algorithms”, SIAM Journal of Control 9 (1971) 529–542.

    Article  MathSciNet  Google Scholar 

  4. H. Hermes, “Calculus of set valued functions and control”, Journal of Mathematics and Mechanics 18 (1968) 47–59.

    MATH  MathSciNet  Google Scholar 

  5. W.W. Hogan, “Point-to-set maps in mathematical programming”, SIAM Review 15 (1973) 591–603.

    Article  MATH  MathSciNet  Google Scholar 

  6. P. Huard, “Optimization algorithms and point-to-set maps”, Mathematical Programming 8 (1975) 308–331.

    Article  MATH  MathSciNet  Google Scholar 

  7. P. Huard, Cours de 3eme cycle, lere partie, elements theoriques, Université de Lille I (France) Janvier 1972.

    Google Scholar 

  8. P. Huard, Cours de 3eme cycle, 2eme partie, algorithmes generaux, Université de Lille I (France) Janvier 1972.

    Google Scholar 

  9. S. Kakutani, “A generalization of Brouwer’s fixed point theorem”, Duke Mathematical Journal 8 (1941) 457–459.

    Article  MATH  MathSciNet  Google Scholar 

  10. D.G. Luenberger, Introduction to linear and nonlinear programming (Addison-Wesley, Reading, MA, 1973).

    MATH  Google Scholar 

  11. G.G.L. Meyer, “Algorithm model for penalty functions-type iterative procedures”, Journal of Computer and System Sciences 9 (1974) 20–30.

    MATH  MathSciNet  Google Scholar 

  12. G.G.L. Meyer, “A canonical structure for iterative procedures”, Journal of Mathematical Analysis and Applications 52 (1975) 120–128.

    Article  MATH  MathSciNet  Google Scholar 

  13. G.G.L. Meyer, “A systematic approach to the synthesis of algorithms”, Numerische Mathematik 24 (1975) 277–289.

    Article  MATH  MathSciNet  Google Scholar 

  14. G.G.L. Meyer, “Conditions de convergence pour les algorithmes iteratifs monotones, autonomes et non deterministes”, Revue Francaise d’Automatique, Informatique et Recherche Operationnelle, Analyse Numerique 11 (1977) 61–74.

    MATH  Google Scholar 

  15. G.G.L. Meyer, “Convergence conditions for a type of algorithm model”, SIAM Journal on Control and Optimization 15 (1977) 779–784.

    Article  MATH  MathSciNet  Google Scholar 

  16. G.G.L. Meyer and E. Polak, “Abstract models for the synthesis of optimization algorithms”, SIAM Journal of Control 9 (1971) 547–560.

    Article  MATH  MathSciNet  Google Scholar 

  17. G.G.L. Meyer and R.C. Raup, “On the structure of cluster point sets of iteratively generated sequences”, Journal of Optimization Theory and Applications, to be published.

    Google Scholar 

  18. R.R. Meyer, “The validity of a family of optimization methods”, SIAM Journal on Control 8 (1979) 41–54.

    Article  Google Scholar 

  19. R.R. Meyer, “Sufficient conditions for the convergence of monotonic mathematical programming algorithms, Technical Report 220, Computer Science Department, University of Wisconsin, Madison, WI.

    Google Scholar 

  20. E. Michael, “Topologies on spaces of subsets”, Transactions American Mathematical Society 71 (1951) 152–182.

    Article  MATH  MathSciNet  Google Scholar 

  21. A.M. Ostrowski, “Solution of equations and systems of equations (Academic Press, New York, 1966).

    MATH  Google Scholar 

  22. E. Polak, “On the convergence of optimization algorithms”, Revue Francaise d’Automatique, Informatique et Recherche Operationnelle 16 (1969) 17–34.

    MathSciNet  Google Scholar 

  23. E. Polak, Computational methods in optimization: a unified approach (Academic Press, New York, 1971).

    Google Scholar 

  24. B.T. Polyak, “Gradient methods for the minimization of functionals”, U.S.S.R. Computational Mathematics and Mathematical Physics 3 (1963) 864–878.

    Article  Google Scholar 

  25. W.I. Zangwill, “Convergence conditions for nonlinear programming algorithms”, Working Paper No. 197, Center for Research in Management Science, University of California, Berkeley, November 1966.

    Google Scholar 

  26. W.I. Zangwill, Nonlinear programming: a unified approach (Prentice Hall, Englewood Cliffs, NJ, 1969).

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. The Johns Hopkins University, Baltimore, Maryland, USA

    Gerard G. L. Meyer

Authors
  1. Gerard G. L. Meyer
    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

Meyer, G.G.L. (1979). Asymptotic properties of sequences iteratively generated by point-to-set maps. In: Huard, P. (eds) Point-to-Set Maps and Mathematical Programming. Mathematical Programming Studies, vol 10. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0120849

Download citation

  • DOI: https://doi.org/10.1007/BFb0120849

  • 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