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Sufficiency and Duality for Nonsmooth Interval-Valued Optimization Problems via Generalized Invex-Infine Functions

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Abstract

In this paper, a new concept of generalized-affineness type of functions is introduced. This class of functions is more general than some of the corresponding ones discussed in Chuong (Nonlinear Anal Theory Methods Appl 75:5044–5052, 2018), Sach et al. (J Global Optim 27:51–81, 2003) and Nobakhtian (Comput Math Appl 51:1385–1394, 2006). These concepts are used to discuss the sufficient optimality conditions for the interval-valued programming problem in terms of the limiting/Mordukhovich subdifferential of locally Lipschitz functions. Furthermore, two types of dual problems, namely Mond–Weir type and mixed type duals are formulated for an interval-valued programming problem and usual duality theorems are derived. Our results improve and generalize the results appeared in Kummari and Ahmad (UPB Sci Bull Ser A 82(1):45–54, 2020).

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References

  1. Chuong, T.D.: L-invex-infine functions and applications. Nonlinear Anal. Theory Methods Appl. 75, 5044–5052 (2018)

    Article  MathSciNet  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  3. Ahmad, I., Jayswal, A., Banerjee, J.: On interval-valued optimization problems with generalized invex functions. J. Inequal. Appl. 2013, 313 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  4. Ahmad, I., Kummari, K., Al-Homidan, S.: Sufficiency and duality for interval-valued optimization problems with vanishing constraints using weak constraint qualification. Int. J. Anal. Appl. 18(5), 784–798 (2020)

    Google Scholar 

  5. Kummari, K., Ahmad, I.: Sufficient optimality conditions and duality for nonsmooth interval-valued optimization problems via L-invex-infine functions. U.P.B. Sci. Bull. Ser. A 82(1), 45–54 (2020)

  6. Kong, X., Zhang, Y., Yu, G.: Optimality and duality in set-valued optimization utilizing limit sets. Open Math. 16, 1128–1139 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  7. Ruiz-Garzón, G., Osuna-Gómez, R., Rufián-Lizana, A., Hernández-Jiménez, B.: Optimality and duality on Riemannian manifolds. Taiwan. J. Math. 22(5), 1245–1259 (2018)

  8. Ahmad, I., Jayswal, A., Al-Homidan, S., Banerjee, J.: Sufficiency and duality in interval-valued variational programming. Neural Comput. Appl. 31, 4423–4433 (2019)

    Article  Google Scholar 

  9. Luu, D.V., Mai, T.T.: Optimality and duality in constrained interval-valued optimization. 4OR-Q. J. Oper. Res. 16(3), 311–337 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  10. Wu, H.: Solving the interval-valued optimization problems based on the concept of null set. J. Ind. Manag. Optim. 14(3), 1157–1178 (2018)

  11. Mordukhovich, B.S.: Maximum principle in problems of time optimal control with nonsmooth constraints. J. Appl. Math. Mech. 40, 960–969 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  12. Chuong, T.D., Kim, D.S.: Optimality conditions and duality in nonsmooth multiobjective optimization problems. Ann. Oper. Res. 217, 117–136 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  13. Khanh, P.D., Yao, J., Yen, N.D.: The Mordukhovich subdifferentials and directions of descent. J. Optim. Theory Appl. 172, 518–534 (2017)

  14. Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, I: Basic Theory. Springer, Berlin (2006)

    Book  Google Scholar 

  15. Mordukhovich, B.S., Shao, Y.H.: Nonsmooth sequential analysis in Asplund spaces. Trans. Am. Math. Soc. 348(4), 1235–1280 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  16. Roshchina, V.: Mordukhovich subdifferential of pointwise minimum of approximate convex functions. Optim. Method Softw. 25, 129–141 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  17. Sach, P.H., LEE, G.M., Kim, D.S.: Infine functions, nonsmooth alternative theorems and vector optimization problems. J. Global Optim. 27, 51–81 (2003)

  18. Nobakhtian, S.: Infine functions and nonsmooth multiobjective optimization problems. Comput. Math. Appl. 51, 1385–1394 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  19. Sun, Y., Wang, L.: Optimality conditions and duality in nondifferentiable interval-valued programming. J. Ind. Manag. Optim. 9, 131–142 (2013)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors are highly thankful to anonymous referees for their valuable suggestions/comments that helped to improve this article in its present form.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran, 31261, Saudi Arabia

    Izhar Ahmad & S. Al-Homidan

  2. Department of Mathematics, School of Science, GITAM, Hyderabad Campus, Hyderabad, 502329, India

    Krishna Kummari

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  1. Izhar Ahmad
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  2. Krishna Kummari
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  3. S. Al-Homidan
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Ahmad, I., Kummari, K. & Al-Homidan, S. Sufficiency and Duality for Nonsmooth Interval-Valued Optimization Problems via Generalized Invex-Infine Functions. J. Oper. Res. Soc. China 11, 505–527 (2023). https://doi.org/10.1007/s40305-021-00381-6

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  • DOI: https://doi.org/10.1007/s40305-021-00381-6

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