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Painlevé-Kuratowski convergence of minimal solutions for set-valued optimization problems via improvement sets

Abstract

The aim of this paper is to explore the stability of (weak)-minimal solutions for set-valued optimization problems via improvement sets. Firstly, the optimality and closedness of solution sets for the set-valued optimization problem under the upper order relation are discussed. Then, a new convergence concept for set-valued mapping sequences is introduced, and some properties of the set-valued mapping sequences are shown under the new convergence assumption. Moreover, by means of upper level sets, Painlevé-Kuratowski convergences of (weak) E-u-solutions to set-valued optimization problems with respect to the perturbations of feasible sets and objective mappings are established under mild conditions. The order that we use to establish the result depends on the improvement set, which is not necessarily a cone order. Our results can be seen as the extension of the related work established recently in this field.

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References

  1. Arrow, K.J., Debreu, G.: Existence of an equilibrium for a comprtitive economy. Econometrica 22, 265–290 (1954)

    MathSciNet  Article  Google Scholar 

  2. Avriel, M., Diewert, W.E., Schaible, S., Zang, I.: Generalized Concavity. Plenum Press, New York (1998)

    MATH  Google Scholar 

  3. Alonso, M., Rodríguez-Marín, L.: Set-relations and optimality conditions in set-valued mappings. Nonlinear Anal. 63, 1167–1179 (2005)

    MathSciNet  Article  Google Scholar 

  4. Anh, L.Q., Duoc, P.T., Tam, T.N.: On the stability of approximate solutions to set-valued equilibrium problems. Optimization 69, 1583–1599 (2019)

    MathSciNet  Article  Google Scholar 

  5. Anh, L.Q., Duy, T.Q., Hien, D.V., Kuroiwa, D., Petrot, N.: Convergence of solution to set optimization problems with the set less order relation. J. Optim. Theory Appl. 185, 416–432 (2020)

    MathSciNet  Article  Google Scholar 

  6. Berge, C.: Topological Spaces. Oliver and Boyd, London (1963)

    MATH  Google Scholar 

  7. Bonnisseau, J.M., Cornet, B.: Existence of marginal cost pricing equilibria in an economy with several nonconvex firms. Econometrica 58, 661–682 (1990)

    MathSciNet  Article  Google Scholar 

  8. Beer, G.: Topologies on Closed and Closed Convex Sets. Springer, Berlin (1993)

    Book  Google Scholar 

  9. Chicco, M., Mignanego, F., Pusillo, L., Tijs, S.: Vector optimization problems via improvement sets. J. Optim. Theory Appl. 150, 516–529 (2011)

    MathSciNet  Article  Google Scholar 

  10. Dhigra, M., Lalitha, C.S.: Set optimization using improvement sets. Yugosl J. Oper. Res. 2, 153–167 (2017)

    MathSciNet  Article  Google Scholar 

  11. Fang, Z.M., Li, S.J.: Painlevé-Kuratowski convergence of the solution sets to perturbed generalized systems. Acta. Math. Appl. Sin-E. 28, 361–370 (2012)

    Article  Google Scholar 

  12. Gale, D.: The law of supply and demand. Math. Scand. 3, 155–169 (1955)

    MathSciNet  Article  Google Scholar 

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

    MATH  Google Scholar 

  14. Gutiérrez, C., Jiménez, B., Novo, V.: Improvement sets and vector optimization. Eur. J. Oper. Res. 223, 304–311 (2012)

    MathSciNet  Article  Google Scholar 

  15. Han, Y., Huang, N.J.: Existence and connectedness of solutions for generalized vector quasi-equilibrium problems. J. Optim. Theory Appl. 179, 65–85 (2016)

    MathSciNet  Article  Google Scholar 

  16. Han, Y., Huang, N.J.: Well-posedness and stability of solutions for set optimization problems. Optimization 66, 17–33 (2017)

    MathSciNet  Article  Google Scholar 

  17. Han, Y., Zhang, K., Huang, N.J.: The stability and extended well-posedness of the solution sets for set optimization problems via the Painlevé-Kuratowski convergence. Math. Meth. Oper. Res. 91, 175–196 (2020)

    Article  Google Scholar 

  18. Han, Y.: Painlevé-Kuratowski convergences of the solution sets for set optimization problems with cone-quasiconnectedness. Optimization (2020). https://doi.org/10.1080/02331934.2020.1857756

    Article  Google Scholar 

  19. Jeyakumar, V.: A generalization of a minimax theorem of Fan via a theorem of the alternative. J. Optim. Theory Appl. 48, 525–533 (1986)

    MathSciNet  Article  Google Scholar 

  20. Kuroiwa, D.: Existence theorems of set optimization with set-valued maps. J. Inf. Optim. Sci. 24, 73–84 (2003)

    MathSciNet  MATH  Google Scholar 

  21. Khoshkhabar-amiranloo, S.: Stability of minimal solutions to parametric set optimization problem. Appl. Anal. 97, 2510–2522 (2018)

    MathSciNet  Article  Google Scholar 

  22. Lalitha, C.S., Chatterjee, P.: Stability and scalarization of weak efficient, efficient and Henig proper efficient sets using generalized quasiconvexities. J. Optim. Theory Appl. 155, 941–961 (2012)

    MathSciNet  Article  Google Scholar 

  23. Lalitha, C.S., Chatterjee, P.: Stability and scalarization in vector optimization using improvement sets. J. Optim. Theory Appl. 166, 825–843 (2015)

    MathSciNet  Article  Google Scholar 

  24. Li, X.B., Lin, Z., Peng, Z.Y.: Convergence for vector optimization problems with variable ordering structure. Optimization 65, 1615–1627 (2016)

    MathSciNet  Article  Google Scholar 

  25. Mao, J.Y., Wang, S.H., Han, Y.: The stability of the solution sets for set optimization problems via improvement sets. Optimization 68, 2171–2193 (2019)

    MathSciNet  Article  Google Scholar 

  26. Peng, Z.Y., Peng, J.W., Long, X.J., Yao, J.C.: On the stability of solutions for semi-infinite vector optimization problems. J. Global Optim. 70, 55–69 (2018)

    MathSciNet  Article  Google Scholar 

  27. Peng, Z.Y., Li, X.B., Long, X.J., Fan, X.D.: Painlevé-Kuratowski stability of approximate efficient solutions for perturbed semi-infinite vector optimization problem. Optim. Lett. 12, 1339–1356 (2018)

    MathSciNet  Article  Google Scholar 

  28. Peng, Z.Y., Wang, Z.Y., Yang, X.M.: Connectedness of solution sets for weak generalized symmetric Ky Fan inequality problems via addition-invariant sets. J. Optim. Theory Appl. 185, 188–206 (2020)

    MathSciNet  Article  Google Scholar 

  29. Peng, Z.Y., Wang, J.J., Long, X.J., Liu, F.P.: Painlevé-Kuratowski convergence of solutions for perturbed symmetric set-valued quasi-equilibrium problem via improvement sets. Asia Pac. J. Oper. Res. 37(04), 2040003 (2020)

  30. Mishra, S.K., Wang, S.Y., Lai, K.K.: Optimality and duality for a multi-objective programming problem involving generalized d-type-I and related n-set functions. Eur. J. Oper. Res. 173, 405–418 (2006)

    MathSciNet  Article  Google Scholar 

  31. Mishra, S.K., Wang, S.Y., Lai, K.K.: Gap function for set-valued vector variational-like inequalities. J. Optimiz. Theory App. 138, 77–84 (2008)

    MathSciNet  Article  Google Scholar 

  32. Mishra, S.K., Upadhyay, B.B., An, L.T.H.: Lagrange multiplier characterizations of solution sets of constrained nonsmooth pseudolinear optimization problems. J. Optimiz. Theory App. 160, 763–777 (2014)

    MathSciNet  Article  Google Scholar 

  33. Mishra, S.K., Jaiswal, M.: Optimality conditions and duality for semi-infinite programming problem with equilibrium constraints. Numer. Func. Anal. Opt. 36, 460–480 (2015)

    MathSciNet  Article  Google Scholar 

  34. Tanaka, T.: Generalized quasiconvexities, cone saddle points and minimax theorems for vector valued functions. J. Optim. Theory Appl. 81, 355–377 (1994)

    MathSciNet  Article  Google Scholar 

  35. Wang, J.J., Peng, Z.Y., Lin, Z., Zhou, D.Q.: On the stability of solutions for the generalized vector quasi-equilibrium problems via free-disposal set. J. Ind. Manag. Optim. 17, 869–887 (2021)

    MathSciNet  Article  Google Scholar 

  36. Yu, P.L.: Cone convexity, cone extreme points and nondominated solutions in decision problems with multiobjectives. J. Optim. Theory Appl. 14, 319–377 (1974)

    MathSciNet  Article  Google Scholar 

  37. Zhao, K.Q., Yang, X.M.: A unified stability result with perturbations in vector optimization. Optim. Lett. 7, 1913–1919 (2013)

    MathSciNet  Article  Google Scholar 

  38. Zhao, Y., Peng, Z.Y., Yang, X.M.: Semicontinuity and convergence for vector optimization problems with approximate equilibrium constraints. Optimization 65, 1397–1415 (2016)

    MathSciNet  Article  Google Scholar 

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Acknowledgements

The work of the first author was completed during his visit to the Shenzhen Research Institute of Big Data, Chinese University of Hong Kong, Shenzhen, China, to which he is grateful to the hospitality received.

Funding

The first author was partially supported by the Chongqing Natural Science Foundation (cstc2021jcyj-msxmX0080), the Postgraduate Research and Innovation Project of Chongqing (2021S0061)” of the second author. the Group Building Scientific Innovation Project for Universities in Chongqing (CXQT21021) and the Education Committee Project Foundation of Bayu Scholar. The study of Yun-Bin Zhao was supported by the National Natural Science Foundation of China (NSFC) under Grant 12071307.

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Correspondence to Zai-Yun Peng.

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Peng, ZY., Chen, XJ., Zhao, YB. et al. Painlevé-Kuratowski convergence of minimal solutions for set-valued optimization problems via improvement sets. J Glob Optim (2022). https://doi.org/10.1007/s10898-022-01166-8

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  • DOI: https://doi.org/10.1007/s10898-022-01166-8

Keywords

  • Set-valued optimization problem
  • Hausdorff K-convergence
  • Painlevé-Kuratowski convergence
  • Improvement sets

Mathematics Subject Classification

  • 49J40
  • 49K40
  • 90C31