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Robust nonsmooth optimality conditions for uncertain multiobjective programs involving stable functions

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Abstract

In this paper, we study and develop robust nonsmooth optimality conditions and duality analysis for an uncertain multiobjective programming problem with constraints ((UCMOP), for brevity). First, we introduce the constraint qualification of the (GRSCQ) type and then establish some robust necessary optimality conditions in terms of the generalized subdifferentials for some types of minima (including robust weakly efficient and robust properly efficient) to such problem involving stable functions. Under suitable assumptions on the pseudo-convexity of objective and constraint functions, robust necessary nonsmooth optimality conditions become robust sufficient optimality conditions. An application of the obtained results for its Wolfe and Mond–Weir types dual problem is presented and some illustrative examples are also provided for our findings.

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References

  1. Aubin, J.-P., Frankowska, H.: Set-Valued Analysis. Birkhauser, Boston (1990)

    Google Scholar 

  2. Ben-Tal, A., Ghaoui, L.E., Nemirovski, A.: Robust Optimization. Princeton Series in Applied Mathematics. Princeton University Press, Princeton (2009)

    Google Scholar 

  3. Ben-Tal, A., Nemirovski, A.: A selected topic in robust convex optimization. Math. Programm. Ser. B. 112, 125–158 (2008)

    Article  MathSciNet  Google Scholar 

  4. Bokrantz, R., Fredriksson, A.: Necessary and sufficient conditions for Pareto efficiency in robust multiobjective optimization. Eur. J. Oper. Res. 262, 682–692 (2017)

    Article  MathSciNet  Google Scholar 

  5. Clarke, F.H.: Optimization and Nonsmooth Analysis. Viley-Interscience, New York (1983)

    Google Scholar 

  6. Clason, C., Khan, A.A., Sama, A., Tammer, C.: Contingent derivatives and regularization for noncoercive inverse problems. Optimization 68(7), 1337–1364 (2019)

    Article  MathSciNet  Google Scholar 

  7. Chen, J., Kobis, E., Yao, J.-C.: Optimality conditions and duality for robust nonsmooth multiobjective optimization problems with constraints. J. Optim. Theory Appl. 181, 411–436 (2019)

    Article  MathSciNet  Google Scholar 

  8. Chuong, T.D.: Optimality and duality for robust multiobjective optimization problems. Nonlinear Anal. 134, 127–143 (2016)

    Article  MathSciNet  Google Scholar 

  9. Duy, T.Q.: Robust efficiency and well-posedness in uncertain vector optimization problems. Optimization 7(4), 937–955 (2023). https://doi.org/10.1080/02331934.2022.2031189

    Article  MathSciNet  Google Scholar 

  10. Fliege, J., Werner, R.: Robust multiobjective optimization & applications in portfolio optimization. Eur. J. Oper. Res. 234, 422–433 (2014)

    Article  MathSciNet  Google Scholar 

  11. Ide, J., Kobis, E.: Concepts of efficiency for uncertain multiobjective problems based on set order relations. Math. Meth. Oper. Res. 80, 99–127 (2014)

    Article  Google Scholar 

  12. Jiménez, B., Novo, V.: First order optimality conditions in vector optimization involving stable functions. Optimization 57(3), 449–471 (2008)

    Article  MathSciNet  Google Scholar 

  13. Jeyakumar, V., Lee, G.M., Li, G.: Characterizing robust solutions sets convex programs under data uncertainty. J. Optim. Theory Appl. 64, 407–435 (2015)

    Article  MathSciNet  Google Scholar 

  14. Jeyakumar, V., Li, G.: Strong duality in robust convex programming: complete characterizations. SIAM J. Optim. 20, 3384–3407 (2010)

    Article  MathSciNet  Google Scholar 

  15. Jeyakumar, V., Li, G., Lee, G.M.: Robust duality for generalized convex programming problems under data uncertainty. Nonlinear Anal. 75, 1362–1373 (2012)

    Article  MathSciNet  Google Scholar 

  16. Gopfert, A., Riahi, H., Tammer, C., Zalinescu, C.: Variational Methods in Partially Ordered Spaces. Springer, New York (2003)

    Google Scholar 

  17. Giorgi, G., Guerraggio, A.: On the notion of tangent cone in mathematical programming. Optimization 25, 11–23 (1992)

    Article  MathSciNet  Google Scholar 

  18. Gunawan, S., Azarm, S.: Multiobjective robust optimization using a sensitivity region concept. Struct. Multidiscrip. Optimization. 29, 50–60 (2005)

    Article  Google Scholar 

  19. Klamroth, K., Kobis, E., Schobel, A., Tammer, C.: A unified approach to uncertain optimization. Eur. J. Oper. Res. 260, 403–420 (2017)

    Article  MathSciNet  Google Scholar 

  20. Klamroth, K., Kobis, E., Schobel, A., Tammer, C.: A unified approach for different concepts of robustness and stochastic programming via nonlinear scalarizing functionals. Optimization 62(5), 649–671 (2013)

    Article  MathSciNet  Google Scholar 

  21. Kim, M.H.: Duality theorem and vector saddle point theorem for robust multiobjective optimization problems. Commun. Korean Math. Soc. 28, 597–602 (2013)

    Article  MathSciNet  Google Scholar 

  22. Lee, J.H., Lee, G.M.: On optimality conditions and duality theorems for robust semi-infinite multiobjective optimization problems. Ann. Oper. Res. 269, 419–438 (2018). https://doi.org/10.1007/s10479-016-2363-5

    Article  MathSciNet  Google Scholar 

  23. Lee, G.M., Lee, J.H.: On nonsmooth optimality theorems for robust multiobjective optimization problems. J. Nonlinear Convex Anal. 16, 2039–2052 (2015)

    MathSciNet  Google Scholar 

  24. Luc, D.T.: Theory of vector optimization. Lecture Notes in Economics and Mathematical Systems, Vol. 39. Springer, Berlin (1989). https://doi.org/10.1007/978-3-642-50280-4

  25. Luc, D.T.: Contingent derivatives of set-valued maps and applications to vector optimization. Math. Program. 50, 99–111 (1991)

    Article  MathSciNet  Google Scholar 

  26. Mond, M., Weir, T.: Generallized Concavity and Duality. Generallized Concavity in Optimization and Economics. Academic Press, New York (1981)

    Google Scholar 

  27. Rodríguez - Marín, L., Sama, M.: About Contingent epiderivatives: J. Math. Anal. Appl. 327, 745–762 (2007)

  28. Rodríguez - Marín, L., Sama, M.: Variational characterization of the contingent epiderivative: J. Math. Anal. Appl. 335, 1374–1382 (2007)

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

    Book  Google Scholar 

  30. Su, T.V.: Optimality conditions for vector equilibrium problems in terms of contingent epiderivatives. Numer. Funct. Anal. Optim. 37, 640–665 (2016)

    Article  MathSciNet  Google Scholar 

  31. Suneja, S.K., Khurana, S., Bhatia, M.: Optimality and duality in vector optimization involving generalized type I functions over cones. J. Global Optim. 49, 23–35 (2011)

    Article  MathSciNet  Google Scholar 

  32. Thuy, N.T.T., Su, T.V.: Robust optimality conditions and duality for nonsmooth multiobjective fractional semi-infinite programming problems with uncertain data. Optimization 72(7), 1745–1775 (2023). https://doi.org/10.1080/02331934.2022.2038154

    Article  MathSciNet  Google Scholar 

  33. Wang, J., Li, S.-J., Chen, C.-R.: Robust nonsmooth optimality conditions for multiobjective optimization problems with infinitely many uncertain constraints. Optimization 72(8), 2039–2067 (2023). https://doi.org/10.1080/02331934.2022.2046740

    Article  MathSciNet  Google Scholar 

  34. Wei, H.-Z., Chen, C.-R., Li, S.-J.: Necessary optimality conditions for nonsmooth robust optimization problems. Optimization 71(7), 1817–1837 (2022). https://doi.org/10.1080/02331934.2020.1836636

    Article  MathSciNet  Google Scholar 

  35. Wolfe, P.: A duality theorem for nonlinear programming. Q. J. Appl. Math. 19, 239–244 (1961)

    Article  Google Scholar 

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Acknowledgements

The author is grateful to the anonymous referee for their valuable comments and suggestions, which helped to improve the quality of the paper.

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  1. Faculty of Mathematics, The University of Danang - University of Science and Education, Da Nang, 550000, Vietnam

    Tran Van Su

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  1. Tran Van Su
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Van Su, T. Robust nonsmooth optimality conditions for uncertain multiobjective programs involving stable functions. Positivity 28, 60 (2024). https://doi.org/10.1007/s11117-024-01077-w

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