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On constrained set-valued optimization problems with \(\rho \)-cone arcwise connectedness

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Abstract

In this paper, we establish the sufficient Karush–Kuhn–Tucker (for short, KKT) conditions of a set-valued optimization problem (P) via contingent epiderivative and \(\rho \)-cone arcwise connectedness assumptions. We also formulate the Mond–Weir type (MWD), Wolfe type (W D), and mixed type (MD) duals of (P) and prove the corresponding weak, strong, and converse duality theorems.

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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, Part A, pp. 160–229. Academic Press, New York (1981)

    Google Scholar 

  2. Aubin, J.P., Frankowska, H.: Set-Valued Analysis. Birhäuser, Boston (1990)

    MATH  Google Scholar 

  3. Avriel, M.: Nonlinear Programming: Theory and Method. Prentice-Hall, Englewood Cliffs (1976)

    MATH  Google Scholar 

  4. Borwein, J.: Multivalued convexity and optimization: a unified approach to inequality and equality constraints. Math. Program. 13(1), 183–199 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  5. Cambini, A., Martein, L., Vlach, M.: Second order tangent sets and optimality conditions. Math. Jpn. 49(3), 451–461 (1999)

    MathSciNet  MATH  Google Scholar 

  6. Corley, H.W.: Existence and \(\rm Lagrangian\) duality for maximizations of set-valued functions. J. Optim. Theory Appl. 54(3), 489–501 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  7. Das, K.: Sufficiency and duality of set-valued fractional programming problems via second-order contingent epiderivative. Yugosl. J. Oper. Res. pp. 1–22 (2021). https://doi.org/10.2298/YJOR210218019D

    Article  Google Scholar 

  8. Das, K., Nahak, C.: Sufficient optimality conditions and duality theorems for set-valued optimization problem under generalized cone convexity. Rend. Circ. Mat. Palermo 63(3), 329–345 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  9. Das, K., Nahak, C.: Sufficiency and duality of set-valued optimization problems via higher-order contingent derivative. J. Adv. Math. Stud. 8(1), 137–151 (2015)

    MathSciNet  MATH  Google Scholar 

  10. Das, K., Nahak, C.: Optimality conditions for approximate quasi efficiency in set-valued equilibrium problems. SeMA J. 73(2), 183–199 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  11. Das, K., Nahak, C.: Set-valued fractional programming problems under generalized cone convexity. Opsearch 53(1), 157–177 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  12. Das, K., Nahak, C.: Approximate quasi efficiency of set-valued optimization problems via weak subdifferential. SeMA J. 74(4), 523–542 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  13. Das, K., Nahak, C.: Optimization problems with difference of set-valued maps under generalized cone convexity. J. Appl. Math. Inform. 35(1–2), 147–163 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  14. Das, K., Nahak, C.: Set-valued minimax programming problems under generalized cone convexity. Rend. Circ. Mat. Palermo 66(3), 361–374 (2017)

    MathSciNet  MATH  Google Scholar 

  15. Das, K., Nahak, C.: Optimality conditions for set-valued minimax fractional programming problems. SeMA J. 77(2), 161–179 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  16. Das, K., Nahak, C.: Optimality conditions for set-valued minimax programming problems via second-order contingent epiderivative. J. Sci. Res. 64(2), 313–321 (2020)

    Google Scholar 

  17. Das, K., Nahak, C.: Sufficiency and duality in set-valued optimization problems under \((p, r)\)-\(\rho \)-\((\eta, \theta )\)-invexity. Acta Univ. Apulensis 62, 93–110 (2020)

    MathSciNet  MATH  Google Scholar 

  18. Das, K., Nahak, C.: Sufficiency and duality of set-valued semi-infinite programming problems under generalized cone convexity. Acta Univ. M. Belii Ser. Math. 2020, 95–111 (2020)

    Google Scholar 

  19. Das, K., Nahak, C.: Parametric set-valued optimization problems under generalized cone convexity. Jnanabha 51(1), 1–11 (2021)

    Article  MATH  Google Scholar 

  20. Das, K., Nahak, C.: Set-valued optimization problems via second-order contingent epiderivative. Yugosl. J. Oper. Res. 31(1), 75–94 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  21. Fu, J.Y., Wang, Y.H.: Arcwise connected cone-convex functions and mathematical programming. J. Optim. Theory Appl. 118(2), 339–352 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  22. Jahn, J., Rauh, R.: Contingent epiderivatives and set-valued optimization. Math. Method Oper. Res. 46(2), 193–211 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  23. Khanh, P.Q., Tung, N.M.: Optimality conditions and duality for nonsmooth vector equilibrium problems with constraints. Optimization 64(7), 1547–1575 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  24. Lalitha, C.S., Dutta, J., Govil, M.G.: Optimality criteria in set-valued optimization. J. Aust. Math. Soc. 75(2), 221–232 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  25. Mishra, S.K., Wang, S.Y., Lai, K.K.: Optimality and duality in nondifferentiable and multiobjective programming under generalized d-invexity. J. Glob. Optim. 29(4), 425–438 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  26. Peng, Z., Xu, Y.: Second-order optimality conditions for cone-subarcwise connected set-valued optimization problems. Acta Math. Appl. Sin. Engl. Ser. 34(1), 183–196 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  27. Qiu, Q., Yang, X.: Connectedness of henig weakly efficient solution set for set-valued optimization problems. J. Optim. Theory Appl. 152(2), 439–449 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  28. Rodríguez-Marín, L., Sama, M.: About contingent epiderivatives. J. Math. Anal. Appl. 327(2), 745–762 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  29. Sach, P.H., Craven, B.D.: Invex multifunctions and duality. Numer. Func. Anal. Opt. 12(5–6), 575–591 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  30. Sach, P.H., Yen, N.D., Craven, B.D.: Generalized invexity and duality theories with multifunctions. Numer. Func. Anal. Opt. 15(1–2), 131–153 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  31. Sheng, B., Liu, S.: Kuhn-Tucker condition and Wolfe duality of preinvex set-valued optimization. Appl. Math. Mech.-Engl. 27, 1655–1664 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  32. Weir, T., Mond, B.: Generalised convexity and duality in multiple objective programming. Bull. Austral. Math. Soc. 39(2), 287–299 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  33. Yihong, X.U., Min, L.I.: Optimality conditions for weakly efficient elements of set-valued optimization with \(\alpha \)-order near cone-arcwise connectedness. J. Syst. Sci. Math. Sci. 36(10), 1721–1729 (2016)

    MathSciNet  MATH  Google Scholar 

  34. Yu, G.: Optimality of global proper efficiency for cone-arcwise connected set-valued optimization using contingent epiderivative. Asia-Pac. J. Oper. Res. 30(03), 1340004 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  35. Yu, G.: Global proper efficiency and vector optimization with cone-arcwise connected set-valued maps. Numer. Algebra Control Optim. 6(1), 35–44 (2016)

    Article  MathSciNet  MATH  Google Scholar 

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The author is very thankful to the editors and referees for their valuable comments which improved the presentation of the paper.

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  1. Department of Mathematics, Taki Government College, Taki, West Bengal, 743429, India

    Koushik Das

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  1. Koushik Das
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Das, K. On constrained set-valued optimization problems with \(\rho \)-cone arcwise connectedness. SeMA 80, 463–478 (2023). https://doi.org/10.1007/s40324-022-00295-0

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