Journal of Global Optimization

, Volume 49, Issue 1, pp 37–47 | Cite as

A class of r-semipreinvex functions and optimality in nonlinear programming



In this paper, a class of functions, named as r-semipreinvex functions, which is generalization of semipreinvex functions and r-preinvex functions, is introduced. Example is given to show that there exists function which is r-semipreinvex function, but is not semipreinvex function. Furthermore, some basic characterizations of r-semipreinvex functions are established. At the same time, some optimality results are obtained in nonlinear programming problems.


Semi-connected sets Semipreinvex functions r-Semipreinvex functions Optimality Nonlinear programming 

Mathematics Subject Classification (2000)

90C25 90C26 90C30 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bazaraa M.S., Shetty C.M.: Nonlinear programming theory and algorithms. Wiley, New York (1979)Google Scholar
  2. 2.
    Schaible S., Ziemba W.T.: Generalized concavity in optimization and economics. Academic press Inc, London (1981)Google Scholar
  3. 3.
    Hanson M.A.: On sufficiency of the Kuhn–Tucker conditions. J. Math. Anal. Appl. 80, 545–550 (1981)CrossRefGoogle Scholar
  4. 4.
    Ben-Israel A., Mond B.: What is invexity? Bull. Aust. Math. Soc. 28, 1–9 (1986)CrossRefGoogle Scholar
  5. 5.
    Weir T., Mond B.: Preinvex functions in multi-objective optimization. J. Math. Anal. Appl. 136, 29–38 (1988)CrossRefGoogle Scholar
  6. 6.
    Weir T., Jeyakumar V.: A class of nonconvex functions and mathematical programming. Bull. Aust. Math. Soc. 38, 177–189 (1988)CrossRefGoogle Scholar
  7. 7.
    Pini R.: Invexity and generalized convexity. Optimization 22, 513–525 (1991)CrossRefGoogle Scholar
  8. 8.
    Lee G.M.: Nonsmooth invexity in multiobjective programming. J. Inf. Optim. Sci. 15, 127–136 (1994)Google Scholar
  9. 9.
    Bhatia D., Kumar P.: Multiobjective control problem with generalized invexity. J. Math. Anal. Appl. 189, 676–692 (1995)CrossRefGoogle Scholar
  10. 10.
    Mohan S.R., Neogy S.K.: On invex sets and preinvex functions. J. Math. Anal. Appl. 189, 901–908 (1995)CrossRefGoogle Scholar
  11. 11.
    Giorgi G., Guerraggio A.: Constraint qualifications in the invex case. J. Inf. Optim. Sci. 19, 373–384 (1998)Google Scholar
  12. 12.
    Reddy L.V., Mukherjee R.N.: Some results on mathematical programming with generalized ratio invexity. J. Math. Anal. Appl. 240, 299–310 (1999)CrossRefGoogle Scholar
  13. 13.
    Yang X.M., Li D.: On properties of preinvex function. J. Math. Anal. Appl. 256, 229–241 (2001)CrossRefGoogle Scholar
  14. 14.
    Yang X.M., Yang X.Q., Teo K.L.: Characterizations and applications of prequasi-invex functions. J. Optim. Theory Appl. 110, 645–668 (2001)CrossRefGoogle Scholar
  15. 15.
    Kim D.S., Schaible S.: Optimality and duality for invex nonsmooth multiobjective programming problems. Optimization 53, 165–176 (2004)CrossRefGoogle Scholar
  16. 16.
    Avriel M.: r-Convex functions. Math. Program. 2, 309–323 (1972)CrossRefGoogle Scholar
  17. 17.
    Antczak T.: r-Preinvexity and r-invexity in mathematical programming. Comput. Math. Appl. 50, 551–566 (2005)CrossRefGoogle Scholar
  18. 18.
    Antczak T.: (p, r)-Invex functions. J. Math. Anal. Appl. 263, 355–379 (2001)CrossRefGoogle Scholar
  19. 19.
    Antczak T.: On (p, r)-invexity-type nonlinear programming Problems. J. Math. Anal. Appl. 264, 382–397 (2001)CrossRefGoogle Scholar
  20. 20.
    Antczak T.: Minimax programming under (p, r)-invexity. Eur. J. Oper. Res. 158, 1–19 (2004)CrossRefGoogle Scholar
  21. 21.
    Antczak T.: A class of B − (p, r)-invex functions and mathematical programming. J. Math. Anal. Appl. 286, 187–206 (2003)CrossRefGoogle Scholar
  22. 22.
    Antczak T.: G-pre-invex functions in mathematical programming. J. Comput. Appl. Math. 26, 1–15 (2007)Google Scholar
  23. 23.
    Yang X.Q., Chen G.Y.: A Class of nonconuex functions and prevariational inequalities. J. Math. Anal. Appl. 169, 359–373 (1992)CrossRefGoogle Scholar
  24. 24.
    Yang X.M., Yang X.Q., Teo K.L.: On properties of semipreinvex functions. Bull. Aust. Math. Soc. 68, 449–459 (2003)CrossRefGoogle Scholar
  25. 25.
    Long X.J., Peng J.W.: Semi-B-preinvex functions. J. Optim. Theory Appl. 131, 301–305 (2006)CrossRefGoogle Scholar
  26. 26.
    Yuan D., Chinchuluun A., Liu X., Pardalos P.M.: Generalized convexities and generalized gradients based on algebraic operations. J. Math. Anal. Appl. 321, 675–690 (2006)CrossRefGoogle Scholar
  27. 27.
    Chinchuluun A., Yuan D., Pardalos P.M.: Optimality Conditions and duality for nondifferentiable multiobjective fractional programming with generalized convexity. Ann. Oper. Res. 154, 133–147 (2007)CrossRefGoogle Scholar
  28. 28.
    Chinchuluun, A., Pardalos, P.M.: Mutiobjective programming problems under generalized convexity. In: Torn, A., Zilinskas, J. (eds.) Models and algorithms for global optimization, pp. 3–20. Springer, Berlin (2007)CrossRefGoogle Scholar
  29. 29.
    Yuan D., Liu X., Chinchuluun A., Pardalos P.M.: Optimality conditions and duality for multiobjective programming involving (C; α; ρ; d)-type I functions. In: Konnov, I.V., Luc, D.T., Rubinov, A.M. (eds) Generalized convexity and related topics, pp. 73–87. Springer, Berlin (2007)Google Scholar

Copyright information

© Springer Science+Business Media, LLC. 2010

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceChongqing Normal UniversityChongqingChina

Personalised recommendations