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Fréchet subdifferentials of efficient point multifunctions in parametric vector optimization

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Abstract

In this paper, we establish new formulae for computing and/or estimating the Fréchet subdifferential of the efficient point multifunction of a parametric vector optimization problem. These formulae are presented in a broad class of conventional vector optimization problems with the presence of geometric, operator and (finite and infinite) functional constraints.

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References

  1. Aubin, J.P.: Contingent derivatives of set-valued maps and existence of solutions to nonlinear inclusions and differential inclusions. In: Nachbin, L. (ed.) Mathematical Analysis and Applications, pp. 159–229. Academic Press, New York (1981)

    Google Scholar 

  2. Bao, T.Q., Mordukhovich, B.S.: Necessary conditions for super minimizers in constrained multiobjective optimization. J. Glob. Optim. 43(4), 533–552 (2009)

    Article  Google Scholar 

  3. Bednarczuk, E.M., Song, W.: Contingent epiderivate and its applications to set-valued maps. Control Cybern. 27, 375–386 (1998)

    Google Scholar 

  4. Canovas, M.J., Lopez, M.A., Mordukhovich, B.S., Parra, J.: Variational analysis in semi-infinite and infinite programming, II: necessary optimality conditions. SIAM J. Optim. 20(6), 2788–2806 (2010)

    Article  Google Scholar 

  5. Chuong, T.D.: Normal subdifferentials of efficient point multifunctions in parametric vector optimization. Optim. Lett. (2012). doi:10.1007/s11590-012-0463-y

  6. Chuong, T.D.: Clarke coderivatives of efficient point multifunctions in parametric vector optimization. Nonlinear Anal. 74, 273–285 (2011)

    Article  Google Scholar 

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

    Article  Google Scholar 

  8. Chuong, T.D., Huy, N.Q., Yao, J.-C.: Pseudo-Lipschitz property of linear semi-infinite vector optimization problems. Eur. J. Oper. Res. 200, 639–644 (2010)

    Article  Google Scholar 

  9. Chuong, T.D., Huy, N.Q., Yao, J.-C.: Subdifferentials of marginal functions in semi-infinite programming. SIAM J. Optim. 20, 1462–1477 (2009)

    Article  Google Scholar 

  10. Chuong, T.D., Yao, J.-C.: Sufficient conditions for pseudo-Lipschitz property in convex semi-infinite vector optimization problems. Nonlinear Anal. 71, 6312–6322 (2009)

    Article  Google Scholar 

  11. Chuong, T.D., Yao, J.-C.: Coderivatives of efficient point multifunctions in parametric vector optimization. Taiwan. J. Math. 13, 1671–1693 (2009)

    Google Scholar 

  12. Chuong, T.D., Yao, J.-C.: Generalized Clarke epiderivatives of parametric vector optimization problems. J. Optim. Theory Appl. 146, 77–94 (2010)

    Article  Google Scholar 

  13. Dinh, N., Goberna, M.A., López, M.A., Son, T.Q.: New Farkas-type constraint qualifications in convex infinite programming. ESAIM Control Optim. Calc. Var. 13, 580–597 (2007)

    Article  Google Scholar 

  14. Dinh, N., Mordukhovich, B.S., Nghia, T.T.A.: Qualification and optimality conditions for DC programs with infinite constraints. Acta Math. Vietnam. 34, 123–153 (2009)

    Google Scholar 

  15. Dinh, N., Mordukhovich, B.S., Nghia, T.T.A.: Subifferentials of value functions and optimality conditions for some classes of DC and bilevel infinite and semi-infinite programs. Math. Program. 123, 101–138 (2010)

    Article  Google Scholar 

  16. Goberna, M.A., López, M.A.: Linear Semi-Infinite Optimization. Wiley, Chichester (1998)

    Google Scholar 

  17. Ioffe, A.D., Tihomirov, V.M.: Theory of extremal problems. North-Holland Publishing Co, Amsterdam-New York (1979)

    Google Scholar 

  18. Huy, N.Q., Mordukhovich, B.S., Yao, J.-C.: Coderivatives of frontier and solution maps in parametric multiobjective optimization. Taiwan. J. Math. 12(8), 2083–2111 (2008)

    Google Scholar 

  19. Kuk, H., Tanino, T., Tanaka, M.: Sensitivity analysis in vector optimization. J. Optim. Theory Appl. 89(3), 713–730 (1996)

    Article  Google Scholar 

  20. Lee, G.M., Huy, N.Q.: On sensitivity analysis in vector optimization. Taiwan. J. Math. 11(3), 945–958 (2007)

    Google Scholar 

  21. Li, C., Ng, K.F., Pong, T.K.: Constraint qualifications for convex inequality systems with applications in constrained optimization. SIAM J. Optim. 19(1), 163–187 (2008)

    Article  Google Scholar 

  22. Li, S., Penot, J.-P., Xue, X.: Codifferential calculus. Set Valued Var. Anal. 19(4), 505–536 (2011)

    Article  Google Scholar 

  23. Mordukhovich, B.S.: Metric approximations and necessary optimality conditions for general classes of nonsmooth extremal problmes. Sov. Math. Dokl. 22, 526–530 (1980)

    Google Scholar 

  24. Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation. I: Basic Theory. Springer, Berlin (2006)

    Google Scholar 

  25. Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation. I: Applications. Springer, Berlin (2006)

    Google Scholar 

  26. Mordukhovich, B.S., Nam, N.M., Yen, N.D.: Subgradients of marginal functions in parametric mathematical programming. Math. Program. Ser. B 116, 369–396 (2009)

    Article  Google Scholar 

  27. Pennot, J.-P.: Compact nets, filters, and relations. J. Optim. Theory Appl. 93, 400–417 (1983)

    Google Scholar 

  28. Reemtsen, R., Rückmann, J.-J. (eds.): Semi-infinite Programming. Nonconvex Optimization and its Applications. Kluwer Academic Publishers, Boston (1998)

    Google Scholar 

  29. Shi, D.S.: Sensitivity analysis in convex vector optimization. J. Optim. Theory Appl. 77(1), 145–159 (1993)

    Article  Google Scholar 

  30. Tanino, T.: Sensitivity analysis in multiobjective optimization. J. Optim. Theory Appl. 56, 479–499 (1988)

    Article  Google Scholar 

  31. Tanino, T.: Stability and sensitivity analysis in convex vector optimization. SIAM J. Control Optim. 26, 521–536 (1988)

    Google Scholar 

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Acknowledgments

The authors would like to thank the referees for valuable comments and suggestions.

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Authors and Affiliations

  1. Department of Mathematics and Applications, Saigon University, 273 An Duong Vuong Street, Ward 3, District 5, Ho Chi Minh City, Vietnam

    Thai Doan Chuong

  2. Center for Fundamental Science, Kaohsiung Medical University, Kaohsiung, 807, Taiwan

    Jen-Chih Yao

  3. Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung, 804, Taiwan

    Jen-Chih Yao

Authors
  1. Thai Doan Chuong
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  2. Jen-Chih Yao
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Corresponding author

Correspondence to Jen-Chih Yao.

Additional information

This work was supported by the National Foundation for Science and Technology Development of Vietnam (NAFOSTED) and the Grant NSC 99-2221-E-037-007-MY3.

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Chuong, T.D., Yao, JC. Fréchet subdifferentials of efficient point multifunctions in parametric vector optimization. J Glob Optim 57, 1229–1243 (2013). https://doi.org/10.1007/s10898-012-0025-y

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  • DOI: https://doi.org/10.1007/s10898-012-0025-y

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