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Piecewise Constant Roughly Convex Functions

  • H.X. Phu
  • N.N. Hai
  • P.T. An
Article
  • 83 Downloads

Abstract

This paper investigates some kinds of roughly convex functions, namely functions having one of the following properties: ρ-convexity (in the sense of Klötzler and Hartwig), δ-convexity and midpoint δ-convexity (in the sense of Hu, Klee, and Larman), γ-convexity and midpoint γ-convexity (in the sense of Phu). Some weaker but equivalent conditions for these kinds of roughly convex functions are stated. In particular, piecewise constant functions \(f:\mathbb{R} \to \mathbb{R}\) satisfying f(x) = f([x]) are considered, where [x] denotes the integer part of the real number x. These functions appear in numerical calculation, when an original function g is replaced by f(x):=g([x]) because of discretization. In the present paper, we answer the question of when and in what sense such a function f is roughly convex.

Generalized convexity rough convexity ρ-convexity δ-convexity midpoint δ-convexity γ-convexity midpoint γ-convexity piecewise constant function 

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References

  1. 1.
    HARTWIG, H., On Generalized Convex Functions, Optimization, Vol. 14, pp. 49–60, 1983.Google Scholar
  2. 2.
    ROBERTS, A. W., and VARBERG, D. E., Convex Functions, Academic Press, New York, NY, 1973.Google Scholar
  3. 3.
    SCHAIBLE, S., and ZIEMBA, W. T., Editors, On Generalized Concavity in Optimization and Economics, Academic Press, New York, NY, 1981.Google Scholar
  4. 4.
    HARTWIG, H., Local Boundedness and Continuity of Generalized Convex Functions, Optimization, Vol. 26, pp. 1–13, 1992.Google Scholar
  5. 5.
    HARTWIG, H., A Note on Roughly Convex Functions, Optimization, Vol. 38, pp. 319–327, 1996.Google Scholar
  6. 6.
    SÖLLNER, B., Eigenschaften γ-grobkonvexer Mengen und Funktionen, Diplomarbeit, Universität Leipzig, Leipzig, Germany, 1991.Google Scholar
  7. 7.
    PHU, H.X., Six Kinds of Roughly Convex Functions, Journal of Optimization Theory and Applications, Vol. 92, pp. 357–375, 1997.Google Scholar
  8. 8.
    HU, T. C., KLEE, V., and LARMAN, D., Optimization of Globally Convex Functions, SIAM Journal on Control and Optimization, Vol. 27, pp. 1026–1047, 1989.Google Scholar
  9. 9.
    PHU, H.X., Some Properties of Globally δ-Convex Functions, Optimization, Vol. 35, pp. 23–41, 1995.Google Scholar
  10. 10.
    PHU, H.X., γ-Subdifferential and γ-Convexity of Functions on the Real Line, Applied Mathematics and Optimization, Vol. 27, pp. 145–160, 1993.Google Scholar
  11. 11.
    PHU, H.X., γ-Subdifferential and γ-Convexity of Functions on a Normed Space, Journal of Optimization Theory and Applications, Vol. 85, pp. 649–676, 1995.Google Scholar
  12. 12.
    PHU, H. X., and HAI, N.N., Some Analytical Properties of γ-Convex Functions on the Real Line, Journal of Optimization Theory and Applications, Vol. 91, pp. 671–694, 1996.Google Scholar
  13. 13.
    KRIPFGANZ, A., Favard's Fonction Penetrante—A Roughly Convex Function, Optimization, Vol. 38, pp. 329–342, 1996.Google Scholar

Copyright information

© Plenum Publishing Corporation 2003

Authors and Affiliations

  • H.X. Phu
    • 1
  • N.N. Hai
    • 2
  • P.T. An
    • 1
  1. 1.Institute of MathematicsHanoiVietnam
  2. 2.Department of Mathematics, Pedagogical CollegeUniversity of HueHueVietnam

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