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Journal of Optimization Theory and Applications

, Volume 91, Issue 3, pp 671–694 | Cite as

Some analytical properties of γ-convex functions on the real line

  • H. X. Phú
  • N. N. Hai
Contributed Papers

Abstract

This paper deals with the analytical properties of γ-convex functions, which are defined as those functions satisfying the inequalityf(x 1 )+f(x 2 )≤f(x1)+f(x2), forx i ∈[x1,x2], |x i x i |=γ, i=1,2, whenever |x1x2|>γ, for some given positive γ. This class contains all convex functions and all periodic functions with period γ. In general, γ-convex functions do not have ideal properties as convex functions. For instance, there exist γ-convex functions which are totally discontinuous or not locally bounded. But γ-convex functions possess so-called conservation properties, meaning good properties which remain true on every bounded interval or even on the entire domain, if only they hold true on an arbitrary closed interval with length γ. It is shown that boundedness, bounded variation, integrability, continuity, and differentiability almost everywhere are conservation properties of γ-convex functions on the real line. However, γ-convex functions have also infection properties, meaning bad properties which propagate to other points, once they appear somewhere (for example, discontinuity). Some equivalent properties of γ-convexity are given. Ways for generating and representing γ-convex functions are described.

Key Words

Generalized convexity globally convex functions γ-convex functions boundedness continuity differentiability integrability 

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References

  1. 1.
    Rockafellar, R. T.,Convex Analysis, Princeton University Press, Princeton, New Jersey, 1970.Google Scholar
  2. 2.
    Roberts, A. W. andVarberg, D. E.,Convex Functions, Academic Press, New York, New York, 1973.Google Scholar
  3. 3.
    Beckenbach, E. F.,Generalized Convex Functions, Bulletin of the American Mathematical Society, Vol. 43, pp. 336–371, 1937.Google Scholar
  4. 4.
    Bector, C. R., Suneja, S. K., andLalitha, C. S.,Generalized B-Vex Functions and Generalized B-Vex Programming, Journal of Optimization Theory and Applications, Vol. 76, pp. 561–576, 1993.CrossRefGoogle Scholar
  5. 5.
    Behringer, F. A.,Discrete and Nondiscrete Quasiconvexlike Functions and Single-Peakedness (Unimodality), Optimization, Vol. 14, pp. 163–181, 1983.Google Scholar
  6. 6.
    Ben-Tal, A.,On Generalized Means and Generalized Convex Functions, Journal of Optimization Theory and Applications, Vol. 21, pp. 1–13, 1977.CrossRefGoogle Scholar
  7. 7.
    Craven, B. D.,Invex Functions and Constrained Local Minima, Bulletin of the Australian Mathematical Society, Vol. 24, pp. 357–366, 1981.Google Scholar
  8. 8.
    Dolecki, S., andKurcyusz, S.,On Φ-Convexity in Extremal Problems, SIAM Journal on Control and Optimization, Vol. 16, pp. 227–300, 1978.CrossRefGoogle Scholar
  9. 9.
    Hanson, M. A.,On Sufficiency of Kuhn-Tucker Conditions, Journal of Mathematical Analysis and Applications, Vol. 80, pp. 545–550, 1981.CrossRefGoogle Scholar
  10. 10.
    Hartwig, H.,On Generalized Convex Functions, Optimization, Vol. 14, pp. 49–60, 1983.Google Scholar
  11. 11.
    Martos, B.,Nonlinear Programming: Theory and Methods, Akademiai Kiado, Budapest, Hungary, 1975.Google Scholar
  12. 12.
    Komlosi, S., Rapcsak, T., andSchaible, S.,Generalized Convexity, Springer Verlag, Berlin, Germany, 1994.Google Scholar
  13. 13.
    Schaible, S., andZiemba, W. T., Editors,On Generalized Concavity in Optimization and Economics, Academic Press, New York, New York, 1981.Google Scholar
  14. 14.
    Hartwig, H.,Generalized Convexities of Lower Semicontinuous Functions, Optimization, Vol. 16, pp. 663–668, 1985.Google Scholar
  15. 15.
    Hartwig, H.,Local Boundedness and Continuity of Generalized Convex Functions, Optimization, Vol. 26, pp. 1–13, 1992.Google Scholar
  16. 16.
    Karamardian, S., andSchaible, S.,Seven Kinds of Monotone Maps, Journal of Optimization Theory and Applications, Vol. 66, pp. 37–46, 1990.CrossRefGoogle Scholar
  17. 17.
    Karamardian, S., Schaible, S., andCrouzeix, J. P.,Characterization of Generalized Monotone Maps, Journal of Optimization Theory and Applications, Vol. 76, pp. 399–413, 1993.CrossRefGoogle Scholar
  18. 18.
    Hu, T. C., Klee, V., andLarman, D.,Optimization of Globally Convex Functions, SIAM Journal on Control and Optimization, Vol. 27, pp. 1026–1047, 1989.CrossRefGoogle Scholar
  19. 19.
    Söllner, B.,Eigenschaften γ-grobkonvexer Mengen und Funktionen Diplomarbeit, Universität Leipzig, 1991.Google Scholar
  20. 20.
    Phú, H. X.,γ-Subdifferential and γ-Convexity of Functions on the Real Line, Applied Mathematics and Optimization, Vol. 27, pp. 145–160, 1993.CrossRefGoogle Scholar
  21. 21.
    Phú, 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
  22. 22.
    Natanson, I. P.,Theorie der Funktionen einer Reellen Veränderlichen Akademie Verlag, Berlin, Germany, 1975.Google Scholar
  23. 23.
    Hobson, E. W.,The Theory of Functions of a Real Variable and the Theory of Fourier Series University Press, Cambridge, England, 1907.Google Scholar

Copyright information

© Plenum Publishing Corporation 1996

Authors and Affiliations

  • H. X. Phú
    • 1
  • N. N. Hai
    • 2
  1. 1.Institute of MathematicsHanoiVietnam
  2. 2.Department of MathematicsUniversity of HueHueVietnam

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