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Optimality Conditions and Duality for Robust Nonsmooth Multiobjective Optimization Problems with Constraints

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Abstract

In this paper, we investigate a robust nonsmooth multiobjective optimization problem related to a multiobjective optimization with data uncertainty. We firstly introduce two kinds of generalized convex functions, which are not necessary to be convex. Robust necessary optimality conditions for weakly robust efficient solutions and properly robust efficient solutions of the problem are established by a generalized alternative theorem and the robust constraint qualification. Further, robust sufficient optimality conditions for weakly robust efficient solutions and properly robust efficient solutions of the problem are also derived. The Mond–Weir-type dual problem and Wolfe-type dual problem are formulated. Finally, we obtain the weak, strong and converse robust duality results between the primal one and its dual problems under the generalized convexity assumptions.

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References

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

    Article  MathSciNet  MATH  Google Scholar 

  2. Ben-Tal, A., Ghaoui, L.E., Nemirovski, A.: Robust Optimization. Princeton series in applied mathematics. Princeton University Press, Princeton (2009)

    MATH  Google Scholar 

  3. Ben-Tal, A., Nemirovski, A.: Robust optimization-methodology and applications. Math. Programm. Ser. B. 92, 453–480 (2002)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  6. 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  MATH  Google Scholar 

  7. 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  MATH  Google Scholar 

  8. Lee, G.M., Yao, J.C.: On solution sets for robust optimization problems. J. Nonlinear Convex Anal. 17, 957–966 (2016)

    MathSciNet  Google Scholar 

  9. Khan, A.A., Tammer, Chr, Zălinescu, C.: Set-Valued Optimization: An Introduction with Applications. Springer, Berlin (2015)

    Book  MATH  Google Scholar 

  10. Gunawan, S., Azarm, S.: Multiobjective robust optimization using a sensitivity region concept. Struct. Multidiscip. Optim. 29, 50–60 (2005)

    Article  Google Scholar 

  11. Deb, K., Gupta, H.: Introducing robustness in multiobjective optimization. Evol. Comput. 14, 463–494 (2006)

    Article  Google Scholar 

  12. Klamroth, K., Köbis, K., Schöbel, A., Tammer, Chr: A unified approach for different concepts of robustness and stochastic programming via nonlinear scalarizing functionals. Optimization 62(5), 649–671 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  13. Kuroiwa, D., Lee, G.M.: On robust multiobjective optimization. Vietnam J. Math. 40, 305–317 (2012)

    MathSciNet  MATH  Google Scholar 

  14. 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  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  16. Goberna, M.A., Jeyakumar, V., Li, G., Vicentre-Perez, J.: Robust solutions of multiobjective linear semi-infinite programs under constraint data uncertainty. SIAM J. Optim. 24, 1402–1419 (2014)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  18. Goberna, M.A., Jeyakumar, V., Li, G., Vicentre-Perez, J.: Robust solutions to multi-objective linear programs with uncertain data. Eur. J. Oper. Res. 242, 730–743 (2015)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  20. Ide, J., Schöbel, A.: Robustness for uncertain multi-objective optimization: a survey and analysis of different concepts. OR Spectr. 38, 235–271 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  21. 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)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  23. 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  MATH  Google Scholar 

  24. Chen, J.W., Cho, Y.J., Kim, J.K., Li, J.: Multiobjective optimization problems with modified objective functions and cone constraints and applications. J. Global Optim. 49, 137–147 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  25. 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  MATH  Google Scholar 

  26. Luc, D.T.: Theory of Vector Optimization, Lecture Notes in Economics and Mathematical Systems. Springer-Verlag, Berlin (1989)

    Google Scholar 

  27. Göpfert, A., Riahi, H., Tammer, Chr, Zălinescu, C.: Variational Methods in Partially Ordered Spaces. Springer, New York (2003)

    MATH  Google Scholar 

  28. Chen, G.Y., Huang, X.X., Yang, X.O.: Vector Optimization: Set-valued and Variational Analysis. Lecture Notes in Economics and Mathematical Systems. Springer-Verlag, Berlin (2005)

    Google Scholar 

  29. Illés, T., Kassay, G.: Theorems of the alternative and optimality conditions for convexlike and general convexlike programming. J. Optim. Theory Appl. 101, 243–257 (1999)

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  31. Göpfert, A., Tammer, C., Riahi, H., Zǎlinescu, C.: Variational Methods in Partially Ordered Spaces. Springer-Verlag, New York (2003)

    MATH  Google Scholar 

  32. Frenk, J.B.G., Kassay, G.: On classes of generalized convex functions, Gordan–Farkas type theorems, and Lagrangian duality. J. Optim. Theory Appl. 102, 315–343 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  33. Giannessi, F.: Constrained Optimization and Image Space Analysis, vol. 1: Separation of Sets and Optimality Conditions. Springer, New York (2005)

    MATH  Google Scholar 

  34. Craven, B.D.: Control and Optimization. Chapman & Hall, London (1995)

    Book  MATH  Google Scholar 

  35. Lee, G.M., Son, P.T.: On nonsmooth optimality theorems for robust optimization problems. Bull. Korean Math. Soc. 51, 287–301 (2014)

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  37. Suneja, S.K., Khurana, S.: Generalized nonsmooth invexity over cones in vector optimization. Eur. J. Oper. Res. 186, 28–40 (2008)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors are grateful to the anonymous referees, the Associate Editor and Professor Giannessi for their constructive comments and valuable suggestions, which have helped to improve the paper. This research was partially supported by the MOST 106-2923-E-039-001-MY3, the Natural Science Foundation of China (11401487), the Basic and Advanced Research Project of Chongqing (cstc2016jcyjA0239), the China Postdoctoral Science Foundation (2015M582512) and the Fundamental Research Funds for the Central Universities.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Southwest University, Chongqing, 400715, China

    Jiawei Chen

  2. Institute of Mathematics, Halle-Wittenberg, 06099, Halle (Saale), Germany

    Elisabeth Köbis

  3. Center for General Education, China Medical University, Taichung, 40402, Taiwan

    Jen-Chih Yao

Authors
  1. Jiawei Chen
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  2. Elisabeth Köbis
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  3. Jen-Chih Yao
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Correspondence to Elisabeth Köbis.

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Akhtar A. Khan.

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Chen, J., Köbis, E. & Yao, JC. Optimality Conditions and Duality for Robust Nonsmooth Multiobjective Optimization Problems with Constraints. J Optim Theory Appl 181, 411–436 (2019). https://doi.org/10.1007/s10957-018-1437-8

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  • DOI: https://doi.org/10.1007/s10957-018-1437-8

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