Journal of Global Optimization

, Volume 61, Issue 2, pp 311–323 | Cite as

Global minimization of the difference of strictly non-positive valued affine ICR functions



In this paper, non-positive valued affine increasing and co-radiant (ICR) functions are defined in the framework of abstract convexity. The basic properties of this class of functions such as support set and subdifferential are presented. As an application, we give optimality conditions for the global minimum of the difference of two strictly non-positive valued affine ICR functions.


Global optimization Abstract convexity Non-positive valued affine ICR function Subdifferential Support set 



The authors are very grateful to the anonymous referees for their useful suggestions on an earlier version of this paper. These suggestions have enabled the authors to improve the paper significantly. This research was supported partially by Graduate University of Advanced Technology and Mahani Mathematical Research Center.


  1. 1.
    Avriel, M., Diewert, W.E., Schaible, S., Zang, I.: Generalized Concavity. Plenum, New York (1988)CrossRefMATHGoogle Scholar
  2. 2.
    Bagirov, A., Rubinov, M.: Global minimization of increasing and positively homogenous functions over the unit simplex. Ann. Oper. Res. 98, 171–187 (2000)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Coelli, T.J., Rao, D.S.P., ÓDonnell, C.J., Battese, G.E.: An Introduction to Efficiency and Productivity Analysis. Springer, Berlin (2005)Google Scholar
  4. 4.
    Diewert, W.E.: Duality approaches to microeconomic theory. In: Arrow, K.J., Intriligator, M.D. (eds.) Handbook of Mathematical Economics, vol. 2, pp. 535–599. North-Holland Publishing Company, Amsterdam (1982)Google Scholar
  5. 5.
    Daryaei, M.H., Mohebi, H.: Abstract convexity of extended real valued ICR functions. Optimization 62(6), 835–855 (2013)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Daryaei, M.H., Mohebi, H.: Characterizations of maximal elements for extended real valued ICR functions. Submitted paperGoogle Scholar
  7. 7.
    Doagooei, A.R., Mohebi, H.: Monotonic analysis over ordered topological vector spaces: IV. J. Glob. Optim. 45, 355–369 (2009)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Doagooei, A.R., Mohebi, H.: Optimization of the difference of ICR functions. Nonlinear Anal. 71, 4493–4499 (2009)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Doagooei, A.R., Mohebi, H.: Optimization of the difference of topical functions. J. Glob. Optim. 57(4), 1349–1358 (2013)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Hiriart, J.B.: From convex to nonconvex minimization: necessary and sufficient conditions for global optimality. In: Nonsmooth Optimization and Related Topics, pp. 219–240. Plenum, New York (1989)Google Scholar
  11. 11.
    Halme, M., Joro, T., Korhonen, P., Salo, S., Wallenius, J.: A value efficiency approach to incorporating preference information in data envelopment analysis. Manag. Sci. 45(1), 103–115 (1999)CrossRefMATHGoogle Scholar
  12. 12.
    Intriligator, M.D.: Mathematical Optimization and Economic Theory. Prentice-Hall, Englewood Cliffs (1971)Google Scholar
  13. 13.
    Martínez-Legaz, J.-E.: Quasiconvex duality theory by generalized conjugation methods. Optimization 19(5), 603–652 (1988)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Mohebi, H., Sadeghi, H.: Monotonic analysis over ordered topological vector spaces: I. Optimization 56(3), 305–321 (2007)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Mohebi, H., Sadeghi, H.: Monotonic analysis over ordered topological vector spaces: II. Optimization 58(2), 241–249 (2009)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Rubinov, A.M.: Abstract Convexity and Global Optimization. Kluwer, Dordrecht (2000)CrossRefMATHGoogle Scholar
  17. 17.
    Rubinov, A.M., Glover, B.M.: Increasing convex-along-rays functions with applications to global optimization. J. Optim. Theory Appl. 102(3), 615–642 (1999)CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Singer, I.: Abstract Convex Analysis. Wiley-Interscience, New York (1997)MATHGoogle Scholar
  19. 19.
    Zarepisheh, M., Soleimani-damaneh, M.: Global variation of outputs with respect to the variation of inputs in performance analysis; generalized RTS. Eur. J. Oper. Res. 186, 786–800 (2008)CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Shahid Bahonar University of KermanKermanIran
  2. 2.Department of MathematicsShahid Bahonar University of KermanKermanIran
  3. 3.Graduate University of Advanced TechnologyKermanIran

Personalised recommendations