Advertisement

Optimization Letters

, Volume 8, Issue 8, pp 2299–2313 | Cite as

Farkas-type results for constrained fractional programming with DC functions

  • Xiang-Kai SunEmail author
  • Yi Chai
  • Jing Zeng
Original Paper

Abstract

In this paper, by using the properties of the epigraph of the conjugate functions, we introduce some closedness conditions and investigate some characterizations of these closedness conditions. Then, by using these closedness conditions, we obtain some Farkas-type results for a constrained fractional programming problem with DC functions. We also show that our results encompass as special cases some programming problems considered in the recent literature.

Keywords

Closedness conditions Farkas-type results Fractional programming DC functions 

Notes

Acknowledgments

The authors would like to thank the Editor and the two anonymous Referees for valuable comments and suggestions, which helped to improve the paper. Research of the first author was partially supported by the National Natural Science Foundation of China (Grant No: 11301570), the Basic and Advanced Research Project of CQ CSTC (Grant No: cstc2013jcyjA00003), the China Postdoctoral Science Foundation funded project (Grant No: 2013M540697) and the Research Fund of Chongqing Technology and Business University (Grant No: 2013-56-03). Research of the second author was partially supported by the National Natural Science Foundation of China (Grant No: 61374135). Research of the third author was partially supported by the Natural Science Foundation Project of CQ CSTC (Grant No: cstc2012jjA00038).

References

  1. 1.
    Dinkelbach, W.: On nonlinear fractional programming. Manag. Sci. 13, 492–498 (1967)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Bernard, J.C., Ferland, J.A.: Convergence of interval-type algorithms for generalized fractional programming. Math. Program. 43, 349–363 (1989)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Yang, X.M., Teo, K.L., Yang, X.Q.: Symmetric duality for a class of nonlinear fractional programming problems. J. Math. Anal. Appl. 271, 7–15 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Yang, X.M., Yang, X.Q., Teo, K.L.: Duality and saddle-point type optimality for generalized nonlinear fractional programming. J. Math. Anal. Appl. 289, 100–109 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Boţ, R.I., Hodrea, I.B., Wanka, G.: Farkas-type results for fractional programming problems. Nonlinear Anal. 67, 1690–1703 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Zhang, X.H., Cheng, C.Z.: Some Farkas-type results for fractional programming with DC functions. Nonlinear Anal. Real World Appl. 10, 1679–1690 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Wang, H.J., Cheng, C.Z.: Duality and Farkas-type results for DC fractional programming with DC constraints. Math. Comput. Model. 53, 1026–1034 (2011)CrossRefzbMATHGoogle Scholar
  8. 8.
    Sun, X.K., Chai, Y.: On robust duality for fractional programming with uncertainty data. Positivity 18, 9–28 (2014)Google Scholar
  9. 9.
    Martinez-Legaz, J.E., Volle, M.: Duality in D. C. programming: the case of several D. C. constraints. J. Math. Anal. Appl. 237, 657–671 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Boţ, R.I., Hodrea, I.B., Wanka, G.: Some new Farkas-type results for inequality system with DC functions. J. Glob. Optim. 39, 595–608 (2007)CrossRefzbMATHGoogle Scholar
  11. 11.
    Boţ, R.I.: Conjugate Duality in Convex Optimization. Springer, Berlin (2010)zbMATHGoogle Scholar
  12. 12.
    Fang, D.H., Li, C., Yang, X.Q.: Stable and total Fenchel duality for DC optimization problems in locally convex spaces. SIAM J. Optim. 21, 730–760 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Sun, X.K., Li, S.J.: Duality and Farkas-type results for extended Ky Fan inequalities with DC functions. Optim. Lett. 7, 499–510 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Sun, X.K., Li, S.J., Zhao, D.: Duality and Farkas-type results for DC infinite programming with inequality constraints. Taiwan. J. Math. 17, 1227–1244 (2013)zbMATHMathSciNetGoogle Scholar
  15. 15.
    Zalinescu, C.: Convex Analysis in General Vector Spaces. World Scientific, London (2002)CrossRefzbMATHGoogle Scholar
  16. 16.
    Burachik, R.S., Jeyakumar, V.: A new geometric condition for Fenchel’s duality in infinite dimensional spaces. Math. Program. 104, 229–233 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Jeyakumar, V., Lee, G.M., Dinh, N.: New sequential Lagrange multiplier conditions characterizing optimality without constraint qualifications for convex programs. SIAM J. Optim. 14, 534–547 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Burachik, R.S., Jeyakumar, V.: A dual condition for the convex subdifferential sum formula with applications. J. Convex Anal. 12, 279–290 (2005)zbMATHMathSciNetGoogle Scholar
  19. 19.
    Boţ, R.I., Grad, S.M., Wanka, G.: New regularity conditions for strong and total Fenchel-Lagrange duality in infinite dimensional spaces. Nonlinear Anal. 69, 323–336 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Boţ, R.I., Grad, S.M., Wanka, G.: On strong and total Lagrange duality for convex optimization problems. J. Math. Anal. Appl. 337, 1315–1325 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Dinh, N., Nghia, T.T.A., Vallet, G.: A closedness condition and its applications to DC programs with convex constraints. Optimization 59, 541–560 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Toland, J.F.: Duality in nonconvex optimization. J. Math. Anal. Appl. 66, 399–415 (1978)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.College of Mathematics and StatisticsChongqing Technology and Business UniversityChongqingChina
  2. 2.College of AutomationChongqing UniversityChongqingChina
  3. 3.State Key Laboratory of Power Transmission Equipment and System Security and New Technology, College of AutomationChongqing UniversityChongqingChina

Personalised recommendations