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Generalized \({\varepsilon }\)-quasi solutions of set optimization problems

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Abstract

We introduce notions of generalized \(\varepsilon \)-quasi solutions to approximate set type solutions of set optimization problems. We study their properties, consistency and limit behavior as approximations to efficient and strict weak efficient solutions. Moreover, we prove an existence result for such solutions and a bound for their asymptotic cone. Finally, we obtain optimality conditions for them.

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References

  1. Ansari, Q.H., Lalitha, C.S., Mehta, M.: Generalized Convexity, Nonsmooth Variational Inequalities, and Nonsmooth Optimization. CRC Press, Boca Raton (2014)

    MATH  Google Scholar 

  2. Attouch, H., Buttazzo, G., Michaille, G.: Variational Analysis in Sobolev and BV Spaces: Applications to PDEs and Optimization. SIAM, Philadelphia (2006)

    Book  Google Scholar 

  3. Attouch, H., Riahi, H.: Stability results for Ekeland’s \(\varepsilon \)-variational principle and cone extremal solutions. Math. Oper. Res. 18, 173–201 (1993)

  4. Aubin, J.-P., Cellina, A.: Differential Inclusions: Set-Valued Maps and Viability Theory. Springer, Berlin (1984)

    Book  Google Scholar 

  5. Auslender, A., Teboulle, M.: Asymptotic Cones and Functions in Optimization and Variational Inequalities. Springer, Berlin (2003)

    MATH  Google Scholar 

  6. Breckner, W.W., Kassay, G.: A systematization of convexity concepts for sets and functions. J. Convex Anal. 4, 109–127 (1997)

    MathSciNet  MATH  Google Scholar 

  7. Dien, P.H.: Locally Lipschitzian set-valued maps and generalized extremal problems with inclusion constraints. Acta Math. Vietnam 8, 109–122 (1983)

    MathSciNet  MATH  Google Scholar 

  8. Dutta, J.: Necessary optimality conditions and saddle points for approximate optimization in Banach spaces. TOP 13, 127–143 (2005)

    Article  MathSciNet  Google Scholar 

  9. Eichfelder, G.: Cone-valued maps in optimization. Appl. Anal. 91, 1831–1846 (2012)

    Article  MathSciNet  Google Scholar 

  10. Ekeland, I.: On the variational principle. J. Math. Anal. Appl. 47, 324–353 (1974)

    Article  MathSciNet  Google Scholar 

  11. Gao, Y., Hou, S.H., Yang, X.M.: Existence and optimality conditions for approximate solutions to vector optimization problems. J. Optim. Theory Appl. 152, 97–120 (2012)

    Article  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  13. Gupta, D., Mehra, A.: Two types of approximate saddle points. Numer. Funct. Anal. Optim. 29, 532–550 (2008)

    Article  MathSciNet  Google Scholar 

  14. Gutiérrez, C., Huerga, L., Köbis, E., Tammer, C.: Approximate solutions of set-valued optimization problems using set-criteria. Appl. Anal. Optim. 3, 501–519 (2017)

    MathSciNet  Google Scholar 

  15. Gutiérrez, C., Jiménez, B., Novo, V.: A generic approach to approximate efficiency and applications to vector optimization with set-valued maps. J. Global Optim. 49, 313–342 (2011)

    Article  MathSciNet  Google Scholar 

  16. Gutiérrez, C., Jiménez, B., Novo, V.: Optimality conditions for quasi-solutions of vector optimization problems. J. Optim. Theory Appl. 167, 796–820 (2015)

    Article  MathSciNet  Google Scholar 

  17. Gutiérrez, C., Jiménez, B., Novo, V., Thibault, L.: Strict approximate solutions in set-valued optimization with applications to the approximate Ekeland variational principle. Nonlinear Anal. 73, 3842–3855 (2010)

    Article  MathSciNet  Google Scholar 

  18. Gutiérrez, C., López, R., Novo, V.: Generalized \(\varepsilon \)-quasi-solutions in multiobjective optimization problems: Existence results and optimality conditions. Nonlinear Anal. 72, 4331–4346 (2010)

    Article  MathSciNet  Google Scholar 

  19. Hernández, E., López, R.: Some useful set-valued maps in set optimization. Optimization 66, 1273–1289 (2017)

    Article  MathSciNet  Google Scholar 

  20. Hernández, E., López, R.: About asymptotic analysis and set optimization. Set Valued Var. Anal. 27, 643–664 (2019)

    Article  MathSciNet  Google Scholar 

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

    Book  Google Scholar 

  22. Khanh, P.Q., Quy, D.N.: Versions of Ekeland’s variational principle involving set perturbations. J. Global Optim. 57, 951–968 (2013)

  23. Kuroiwa, D.: On set-valued optimization. In: Proceedings of the Third World Congress of Nonlinear Analysts, Part 2, Nonlinear Analysis, vol. 47, pp. 1395–1400 (2001)

  24. Kuroiwa, D., Nuriya, T.: A generalized embedding vector space in set optimization. In: Hsu, S.-B., et al. (eds.) Nonlinear Analysis and Convex Analysis, pp. 297–303. Yokohama Publishers, Yokohama (2007)

    Google Scholar 

  25. Lemaire, B.: Approximation in multiobjective optimization. J. Global Optim. 2, 117–132 (1992)

    Article  MathSciNet  Google Scholar 

  26. Loridan, P.: Necessary conditions for \(\varepsilon \)-optimality. Math. Program. Stud. 19, 140–152 (1982)

    Article  MathSciNet  Google Scholar 

  27. Loridan, P.: \(\varepsilon \)-solutions in vector minimization problems. J. Optim. Theory Appl. 43, 265–276 (1984)

    Article  MathSciNet  Google Scholar 

  28. Luderer, B., Minchenko, L., Satsura, T.: Multivalued Analysis and Nonlinear Programming Problems with Perturbations. Springer, Dordrecht (2002)

    Book  Google Scholar 

  29. Minh, N.B., Tan, N.X.: On the continuity of vector convex multivalued functions. Acta Math. Vietnam 27, 13–25 (2002)

    MathSciNet  MATH  Google Scholar 

  30. Qiu, J.H.: Ekeland’s variational principle in Fréchet spaces and the density of extremal points. Stud. Math. 168, 81–94 (2005)

  31. Qiu, J.H.: A generalized Ekeland vector variational principle and its applications in optimization. Nonlinear Anal. 71, 4705–4717 (2009)

    Article  MathSciNet  Google Scholar 

  32. Qiu, J.H., He, F.: Ekeland variational principles for set-valued functions with set perturbations. Optimization 69, 925–960 (2020)

    Article  MathSciNet  Google Scholar 

  33. Qiu, Q., Yang, X.: Some properties of approximate solutions for vector optimization problem with set-valued functions. J. Global Optim. 47, 1–12 (2010)

    Article  MathSciNet  Google Scholar 

  34. Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer, Berlin (1998)

    Book  Google Scholar 

  35. Seto, K., Kuroiwa, D., Popovici, N.: A systematization of convexity and quasiconvexity concepts for set-valued maps defined by l-type and u-type preorder relations. Optimization 67, 1077–1094 (2018)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The authors are very grateful to the anonymous referee for his/her helpful comments and suggestions. This work has been partially supported by projects: PID2020-112491GB-I00 / AEI / 10.13039/501100011033 through the Ministerio de Ciencia e Innovación, Agencia Estatal de Investigación, Spain (Gutiérrez, López), Fondecyt 1181368 through ANID-Chile (López, Martínez) and Apoyo Institucional para el Magíster Académico de la Universidad de Tarapacá, Chile (Martínez).

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Authors and Affiliations

  1. Mathematics Research Institute (IMUVA), University of Valladolid, Paseo de Belén S/N, Campus Miguel Delibes, 47011, Valladolid, Spain

    C. Gutiérrez

  2. Departamento de Matemática, Universidad de Tarapacá, Avenida 18 de Septiembre 2222, Campus Saucache, Casilla 7B, Arica, Chile

    R. López & J. Martínez

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  1. C. Gutiérrez
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  2. R. López
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  3. J. Martínez
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Correspondence to R. López.

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Gutiérrez, C., López, R. & Martínez, J. Generalized \({\varepsilon }\)-quasi solutions of set optimization problems. J Glob Optim 82, 559–576 (2022). https://doi.org/10.1007/s10898-021-01098-9

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  • DOI: https://doi.org/10.1007/s10898-021-01098-9

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