Skip to main content
Log in

Robust optimality, duality and saddle points for multiobjective fractional semi-infinite optimization with uncertain data

  • Original Paper
  • Published:
Optimization Letters Aims and scope Submit manuscript

Abstract

This paper is devoted to the investigation of a class of uncertain multiobjective fractional semi-infinite optimization problems (\(UMFP \), for brevity). We first obtain, by combining robust optimization and scalarization methodologies, necessary and sufficient optimality conditions for robust approximate weakly efficient solutions of (\(UMFP \)). Then, we introduce a Mixed type approximate dual problem for (\(UMFP \)) and investigate their robust approximate duality relationships. Moreover, we obtain some robust approximate weak saddle point theorems for an uncertain multiobjective Lagrangian function related to (\(UMFP \)).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Dinkelbach, W.: On nonlinear fractional programming. Manag. Sci. 1967, 492–498 (1967)

    Article  MathSciNet  Google Scholar 

  2. Egudo, R.: Multiobjective fractional duality. Bull. Austral. Math. Soc. 37, 367–378 (1988)

    Article  MathSciNet  Google Scholar 

  3. Liu, J.C., Yokoyama, K.: \(\epsilon \)-Optimality and duality for multiobjective fractional programming. Comput. Math. Appl. 37, 119–128 (1999)

    Article  MathSciNet  Google Scholar 

  4. 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)

    Article  MathSciNet  Google Scholar 

  5. 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)

    Article  MathSciNet  Google Scholar 

  6. Long, X.J., Huang, N.J., Liu, Z.B.: Optimality conditions, duality and saddle points for nondifferentiable multiobjective fractional programs. J. Ind. Manag. Optim. 4, 287–298 (2008)

    Article  MathSciNet  Google Scholar 

  7. Long, X.J.: Optimality conditions and duality for nondifferentiable multiobjective fractional programming problems with \((C, \alpha, \rho, d)\)-convexity. J. Optim. Theory Appl. 148, 197–208 (2011)

    Article  MathSciNet  Google Scholar 

  8. Verma, R.U.: Weak \(\epsilon \)-efficiency conditions for multiobjective fractional programming. Appl. Math. Comput. 219, 6819–6827 (2013)

    MathSciNet  MATH  Google Scholar 

  9. Antczak, T.: Parametric saddle point criteria in semi-infinite minimax fractional programming problems under \((p, r)\)-invexity. Numer. Funct. Anal. Optim. 36, 1–28 (2015)

    Article  MathSciNet  Google Scholar 

  10. Khanh, P.Q., Tung, L.T.: First- and second-order optimality conditions for multiobjective fractional programming. Top 23, 419–440 (2015)

    Article  MathSciNet  Google Scholar 

  11. Aubry, A., Carotenuto, V., De Maio, A.: New results on generalized fractional programming problems with Toeplitz quadratics. IEEE Signal Process. Lett. 23, 848–852 (2016)

    Article  Google Scholar 

  12. Chuong, T.D.: Nondifferentiable fractional semi-infinite multiobjective optimization problems. Oper. Res. Lett. 44, 260–266 (2016)

    Article  MathSciNet  Google Scholar 

  13. Stancu-Minasian, I.M.: A ninth bibliography of fractional programming. Optimization 68, 2125–2169 (2019)

    Article  MathSciNet  Google Scholar 

  14. Su, T.V., Hang, D.D.: Optimality and duality in nonsmooth multiobjective fractional programming problem with constraints. 4OR-Q. J. Oper. Res. (2021). https://doi.org/10.1007/s10288-020-00470-x

  15. Ben-Tal, A., Ghaoui, L.E., Nemirovski, A.: Robust Optimization. Princeton University Press, Princeton (2009)

    Book  Google Scholar 

  16. Bertsimas, D., Brown, D.B., Caramanis, C.: Theory and applications of robust optimization. SIAM Rev. 53, 464–501 (2011)

    Article  MathSciNet  Google Scholar 

  17. Beck, A., Ben-Tal, A.: Duality in robust optimization: primal worst equals dual best. Oper. Res. Lett. 37, 1–6 (2009)

    Article  MathSciNet  Google Scholar 

  18. Gabrel, V., Murat, C., Thiele, A.: Recent advances in robust optimization: an overview. Eur. J. Oper. Res. 235, 471–483 (2014)

    Article  MathSciNet  Google Scholar 

  19. Lee, J.H., Lee, G.M.: On \(\epsilon \)-solutions for robust fractional optimization problems. J. Inequal. Appl. 2014, 501 (2014)

    Article  MathSciNet  Google Scholar 

  20. Sun, X.K., Chai, Y.: On robust duality for fractional programming with uncertainty data. Positivity. 18, 9–28 (2014)

    Article  MathSciNet  Google Scholar 

  21. Ide, J., Schöbel, A.: Robustness for uncertain multiobjective optimization: a survey and analysis of different concepts. OR Spectrum. 38, 235–271 (2016)

    Article  Google Scholar 

  22. Sun, X.K., Li, X.B., Long, X.J., Peng, Z.Y.: On robust approximate optimal solutions for uncertain convex optimization and applications to multi-objective optimization. Pac. J. Optim. 13, 621–643 (2017)

    MathSciNet  MATH  Google Scholar 

  23. Sun, X.K., Long, X.J., Fu, H.Y., Li, X.B.: Some characterizations of robust optimal solutions for uncertain fractional optimization and applications. J. Ind. Manag. Optim. 13, 803–824 (2017)

    Article  MathSciNet  Google Scholar 

  24. Fakhar, M., Mahyarinia, M.R., Zafarani, J.: On nonsmooth robust multiobjective optimization under generalized convexity with applications to portfolio optimization. Eur. J. Oper. Res. 265, 39–48 (2018)

    Article  MathSciNet  Google Scholar 

  25. Li, X.B., Wang, Q.L., Lin, Z.: Optimality conditions and duality for minimax fractional programming problems with data uncertainty. J. Ind. Manag. Optim. 15, 1133–1151 (2019)

    MathSciNet  MATH  Google Scholar 

  26. Zeng, J., Xu, P., Fu, H.Y.: On robust approximate optimal solutions for fractional semi-infinite optimization with uncertainty data. J. Inequal. Appl. 2019, 45 (2019)

    Article  MathSciNet  Google Scholar 

  27. Sun, X.K., Teo, K.L., Tang, L.P.: Dual approaches to characterize robust optimal solution sets for a class of uncertain optimization problems. J. Optim. Theory Appl. 182, 984–1000 (2019)

    Article  MathSciNet  Google Scholar 

  28. Wei, H.Z., Chen, C.R., Li, S.J.: A unified approach through image space analysis to robustness in uncertain optimization problems. J. Optim. Theory Appl. 184, 466–493 (2020)

    Article  MathSciNet  Google Scholar 

  29. Wei, H.Z., Chen, C.R., Li, S.J.: Robustness characterizations for uncertain optimization problems via image space analysis. J. Optim. Theory Appl. 186, 459–479 (2020)

    Article  MathSciNet  Google Scholar 

  30. Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)

    Book  Google Scholar 

  31. Goberna, M.A., López, M.A.: Linear Semi-infinite Optimization. Wiley, Chichester (1998)

    MATH  Google Scholar 

  32. Rockafellar, R.T.: Extension of Fenchels duality theorem for convex functions. Duke Math. J. 33, 81–89 (1966)

    Article  MathSciNet  Google Scholar 

  33. Fang, D.H., Li, C., Ng, K.F.: Constraint qualifications for extended Farkass lemmas and Lagrangian dualities in convex infinite programming. SIAM J. Optim. 20, 1311–1332 (2009)

    Article  MathSciNet  Google Scholar 

  34. Sun, X.K., Fu, H.Y., Zeng, J.: Robust approximate optimality conditions for uncertain nonsmooth optimization with infinite number of constraints. Mathematics. 7, 12 (2019)

    Article  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the anonymous referees for valuable comments and suggestions, which helped to improve the paper

Funding

This research was supported by the Natural Science Foundation of Chongqing (cstc2020jcyj-msxmX0016), the ARC Discovery Grant (DP190103361), the Education Committee Project Foundation of Chongqing for Bayu Young Scholar, the Project of CTBU (ZDPTTD201908), and the Innovation Project of CTBU (yjscxx2021-112-58).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Xiangkai Sun.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sun, X., Feng, X. & Teo, K.L. Robust optimality, duality and saddle points for multiobjective fractional semi-infinite optimization with uncertain data. Optim Lett 16, 1457–1476 (2022). https://doi.org/10.1007/s11590-021-01785-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11590-021-01785-2

Keywords

Navigation