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Painlevé–Kuratowski stability of approximate efficient solutions for perturbed semi-infinite vector optimization problems

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Abstract

This paper is concerned with the stability of semi-infinite vector optimization problems (SIVOP) under functional perturbations of both objective functions and constraint sets. First, we establish the Berge-lower semicontinuity and Painlevé–Kuratowski convergence of the constraint set mapping. Then, using the obtained results, we obtain sufficient conditions of Painlevé–Kuratowski stability for approximate efficient solution mapping and approximate weakly efficient solution mapping to the (SIVOP). Furthermore, an application to the traffic network equilibrium problems is also given.

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References

  1. Chen, G.Y., Craven, B.D.: Existence and continuity of solutions for vector optimization. J. Optim. Theory Appl. 81, 459–468 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  2. Chen, G.Y., Huang, X.X., Yang, X.Q.: Vector Optimization: Set-Valued and Variational Analysis. Springer, Berlin (2005)

    MATH  Google Scholar 

  3. Lucchetti, R., Revaliski, J. (eds.): Recent Developments in Well-Posed Variarional Problems. Kluwer, Dordrecht (1995)

    MATH  Google Scholar 

  4. Long, X.J., Peng, Z.Y., Wang, X.F.: Characterizations of the solution set for nonconvex semi-infinite programming problems. J. Nonlinear Convex Anal. 17, 251–265 (2016)

    MathSciNet  MATH  Google Scholar 

  5. Mishra, S.K., Jaiswal, M., Le Thi, H.A.: Nonsmooth semi-infinite programming problem using limiting subdifferentials. J. Glob. Optim. 53, 285–296 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  6. Cánovas, M.J., Kruger, A.Y., López, M.A., Parra, J., Théra, M.A.: Calmness modulus of linear semi-infinite programs. SIAM J. Optim. 24, 29–48 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chuong, T.D., Huy, N.Q., Yao, J.C.: Stability of semi-infinite vector optimization problems under functional perturbations. J. Glob. Optim. 45, 583–595 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  8. Huy, N.Q., Yao, J.C.: Semi-infinite optimization under convex function perturbations: Lipschitz stability. J. Optim. Theory Appl. 128, 237–256 (2011)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  10. Huang, X.X.: Stability in vector-valued and set-valued optimization. Math. Methods Oper. Res. 52, 185–195 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  11. Lucchetti, R.E., Miglierina, E.: Stability for convex vector optimization problems. Optimization 53, 517–528 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  12. Lalitha, C.S., Chatterjee, P.: Stability for properly quasiconvex vector optimization problem. J. Optim. Theory Appl. 155, 492–506 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  13. Durea, M.: On the existence and stability of approximate solutions of perturbed vector equilibrium problems. J. Math. Anal. Appl. 333, 1165–1179 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  14. 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  MATH  Google Scholar 

  15. Zhao, Y., Peng, Z.Y., Yang, X.M.: Painlevé–Kuratowski convergences of the solution sets for perturbed generalized systems. J. Nonlinear Convex Anal. 15, 1249–1259 (2014)

    MathSciNet  MATH  Google Scholar 

  16. Peng, Z.Y., Yang, X.M.: Painlevé–Kuratowski convergences of the solution sets for perturbed vector equilibrium problems without monotonicity. Acta Math. Appl. Sin. E. 30, 845–858 (2014)

    Article  MATH  Google Scholar 

  17. Li, X.B., Lin, Z., Wang, Q.L.: Stability of approximate solution mappings for generalized Ky Fan inequality. Top 24, 1–10 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  18. Gong, X.H.: Lower semicontinuity of the efficient solution mapping in semi-infinite vector optimization. J. Syst. Sci. Complex. 28, 1312–1325 (2015)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  20. Luc, D.T.: Theory of Vector Optimization, Lecture Notes in Economics and Mathematical Systems, vol. 319. Springer, Berlin (1989)

    Google Scholar 

  21. Aubin, J.P., Ekeland, I.: Applied Nonlinear Analysis. Wiley, New York (1984)

    MATH  Google Scholar 

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

    MATH  Google Scholar 

  23. Wardrop, J.G.: Some theoretical aspects of road traffic research. In: Proceedings of the Institute of Civil Engineers, Part II, pp. 325–378 (1952)

  24. Chen, G.Y., Yen, N.D.: On the Variational Inequality Model for Network Equilibrium, Internal Report 3.196 (724), Department of Mathematics, University of Pisa (1993)

  25. Yang, X.Q., Goh, C.J.: On vector variational inequalities: application to vector equilibria. J. Optim. Theory Appl. 95, 431–443 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  26. Giannessi, F.: Vector Variational Inequalities and Vector Equilibria, Mathematical Theories. Kluwer, Dordrecht (2000)

    Book  MATH  Google Scholar 

  27. Konnov, I.V.: Vector network equilibrium problems with elastic demands. J. Glob. Optim. 57, 521–531 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  28. Smith, M.J.: The existence, uniqueness and stability of traffic equilibrium. Transp. Res. 138, 295–304 (1979)

    Article  Google Scholar 

  29. De Luca, M.: Generalized quasivariational inequalities and traffic equilibrium problems. In: Giannessi, F., Maugeri, A. (eds.) Variational Inequalities and Network Equilibrium Problems, pp. 45–54. Plenum, New York (1995)

    Chapter  Google Scholar 

  30. Maugeri, A.: Variational and quasivariational inequalities in network flow models: recent developments in theory and algorithms. In: Giannessi, F., Maugeri, A. (eds.) Variational Inequalities and Network Equilibrium Problems, pp. 195–211. Plenum, New York (1995)

    Google Scholar 

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Acknowledgements

The work of the first author was completed during his visit to the Department of Mathematics, University of British Columbia, Kelowna, Canada, to which he is grateful to the hospitality received.

Funding

This research was supported by the National Natural Science Foundation of China (11301571,11471059), the Basic and Advanced Research Project of Chongqing (cstc2015jcyjB00001, cstc2017jcyjAX0382, cstc2016jcyjA0219), the China Postdoctoral Science Foundation funded project (2015M580774,2016T90837), the Program for University Innovation Team of Chongqing (CXTDX201601022, CXTDX201601026) and the Education Committee Project Foundation of Bayu Scholar.

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Authors and Affiliations

  1. College of Mathematics and Statistics, Chongqing JiaoTong University, Chongqing, 400074, People’s Republic of China

    Z. Y. Peng & X. B. Li

  2. College of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing, 400067, People’s Republic of China

    X. J. Long

  3. Department of Mathematics, Bohai University, Jinzhou, 121013, Liaoning, People’s Republic of China

    X. D. Fan

Authors
  1. Z. Y. Peng
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  2. X. B. Li
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  3. X. J. Long
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  4. X. D. Fan
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Correspondence to Z. Y. Peng.

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Peng, Z.Y., Li, X.B., Long, X.J. et al. Painlevé–Kuratowski stability of approximate efficient solutions for perturbed semi-infinite vector optimization problems. Optim Lett 12, 1339–1356 (2018). https://doi.org/10.1007/s11590-017-1175-0

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