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Mekong Delta Natural Science Conference
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Pham, T. H., Nguyen, T. S., & Huynh, H. P. (2025). On generalized second – order asymptotic derivatives and application in parametric vector optimization problems. Dong Thap University Journal of Science, 14(04S), 283-296. https://doi.org/10.52714/dthu.14.04S.2025.1587
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On generalized second – order asymptotic derivatives and application in parametric vector optimization problems
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Abstract
This paper deals with generalized second-order asymptotic derivatives of frontier and solution maps in parametric vector optimization problems. Under some mild conditions, we obtain some formulas for computing generalized second-order asymptotic derivatives of frontier and solution maps, respectively.
Keywords
Parametric vector optimization, generalized second-order asymptotic derivatives, frontier and solution maps, sensitivity analysis
Article Details
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.
References
Aubin, J. P., & Frankowska, H. (1990). Set-valued Analysis, Basel: Birkäuser.
Bonnans, J.F., & Shapiro, A. (2000). Perturbation Analysis of Optimization Problems, New York: Springer.
Brecker, W. W., & Kassay, G. (1997). A systematization of convexity concepts for sets and functions. J. Convex Anal., 4, 109-127. https://www.heldermann-verlag.de/jca/jca04/jca04005.pdf
Chuong, T. D., & Yao, J. C. (2010). Generalized clarke epiderivatives of parametric vector optimization problems. J. Optim. Theory Appl., 147, 77-94. https://doi.org/10.1007/s10957-010-9646-9
Chuong, T. D. (2013). Derivatives of the efficient point multifunction in parametric vector optimization problems. J. Optim. Theory Appl., 156, 247-265. https://doi.org/10.1007/s10957-012-0099-1
Durea, M. (2004). First and second order optimality conditions for set-valued optimization problems, Rend. Circ. Mat. Palermo., 2, 451-468. https://doi.org/10.1007/BF02875738
Kalashnikov, V., Jadamba, B., & Khan, A. A. (2006). First and second-order optimality conditions in set optimization. In: Dempe, S., Kalashnikov, V. (eds.) Optimization with Multivalued Mappings, 265-276.
Khan, A. A, & Tammer, C. (2013). Second-order optimality conditions in set-valued optimization via asymptotic derivatives. Optimization., 62, 743-758. https://doi.org/10.1080/02331934.2012.674948
Khan, A. A, Tammer, C., & Zlinescu, C. ( 2015). Set-Valued Optimization: An Introduction with Applications. Heidelberg: Springer.
Khanh, P. Q, and Tung, N. M. (2016). Second-order conditions for open-cone minimizers and firm minimizers in set-valued optimization subject to mixed constraints. J. Optim. Theory Appl., 171, 45-69. https://doi.org/10.1007/s10957-016-0995-x
Kuk, H., Tanino, T., & Tanaka, M. (1996a). Sensitivity analysis in parametrized convex vector optimization. J. Math. Anal. Appl., 202, 511-522. https://doi.org/10.1006/jmaa.1996.0331
Kuk, H., Tanino, T., & Tanaka, M. (1996b). Sensitivity analysis in vector optimization. J. Optim. Theory Appl., 89, 713-730. https://doi.org/10.1007/BF02275356
Li, S. J., Sun, X. K., & Zhai, J. (2012). Second-order contingent derivatives of set-valued mappings with application to set-valued optimization. Appl. Math. Comput., 218, 6874-6886. https://doi.org/10.1016/j.amc.2011.12.064
Li, S. J., Zhu, S. K., & Li, X. B. (2012). Second order optimality conditions for strict efficiency of constrained set-valued optimization. J. Optim. Theory Appl., 155, 534-557. https://doi.org/10.1007/s10957-012-0076-8
Li, S. J., & Zhai J. (2012). Second-order asymptotic differential properties and optimality conditions for weak vector variational inequalities. Optim. Lett., 6, 503-523. https://doi.org/10.1007/s11590-011-0276-4
Mordukhovich, B. S. (2006). Variational Analysis and Generalized Differentiation. Berlin: Springer, I: Basic Theory.
Penot, J. P. (1998). Second-order conditions for optimization problems with constraints. SIAM J. Control Optim., 37, 303-318. https://doi.org/10.1137/S0363012996311095
Sawaragi, Y., Nakayama, H., & Tanino, T. (1985). Theory of Multiobjective Optimization. New York: Academic Press.
Shi, D. S. (1991). Contingent derivative of the perturbation map in multiobjective optimization. J. Optim. Theory Appl., 70, 385-396. https://doi.org/10.1007/BF00940634
Shi, D. S. (1993). Sensitivity analysis in convex vector optimization. J. Optim. Theory Appl., 77, 145-159. https://doi.org/10.1007/BF00940783
Sun, X. K., & Li, S. J. (2014). Generalized second-order contingent epiderivatives in parametric vector optimization problem. J. Glob. Optim., 58, 351-363. https://doi.org/10.1007/s10898-013-0054-1
Tanino, T. (1988a). Sensitivity analysis in multiobjective optimization. J. Optim. Theory Appl., 56, 479-499. https://doi.org/10.1007/BF00939554
Tanino, T. (1988b). Stability and sensitivity analysis in convex vector optimization. SIAM J. Control Optim., 26, 521-536. https://doi.org/10.1137/0326031
Tung, L. T. (2017a). Second-order radial-asymptotic derivatives and applications in set-valued vector optimization. Pac. J. Optim., 13, 137-153. http://www.yokohamapublishers.jp/online-p/PJO/vol13/pjov13n1p137.pdf
Tung, L. T. (2017b). Variational sets and asymptotic variational sets of proper perturbation map in parametric vector optimization. Positivity., 21, 1647-1673. https://doi.org/10.1007/s11117-017-0491-z
Tung, L. T. (2018). On second-order proto-differentiability of perturbation maps. Set-Valued Var. Anal., 26, 561-579. https://doi.org/10.1007/s11228-016-0397-0
Tung, L. T. (2021). On higher-order proto-differentiability and higher-order asymptotic proto-differentiability of weak perturbation maps in parametric vector optimization. Positivity, 25(2), 579-604. https://doi.org/10.1007/s11117-020-00778-2
Tung, N. M. (2021). Second-order efficient optimality conditions for set-valued vector optimization in terms of asymptotic contingent epiderivatives. RAIRO Oper. Res., 55, 841-860. https://doi.org/10.1051/ro/2021039
Bonnans, J.F., & Shapiro, A. (2000). Perturbation Analysis of Optimization Problems, New York: Springer.
Brecker, W. W., & Kassay, G. (1997). A systematization of convexity concepts for sets and functions. J. Convex Anal., 4, 109-127. https://www.heldermann-verlag.de/jca/jca04/jca04005.pdf
Chuong, T. D., & Yao, J. C. (2010). Generalized clarke epiderivatives of parametric vector optimization problems. J. Optim. Theory Appl., 147, 77-94. https://doi.org/10.1007/s10957-010-9646-9
Chuong, T. D. (2013). Derivatives of the efficient point multifunction in parametric vector optimization problems. J. Optim. Theory Appl., 156, 247-265. https://doi.org/10.1007/s10957-012-0099-1
Durea, M. (2004). First and second order optimality conditions for set-valued optimization problems, Rend. Circ. Mat. Palermo., 2, 451-468. https://doi.org/10.1007/BF02875738
Kalashnikov, V., Jadamba, B., & Khan, A. A. (2006). First and second-order optimality conditions in set optimization. In: Dempe, S., Kalashnikov, V. (eds.) Optimization with Multivalued Mappings, 265-276.
Khan, A. A, & Tammer, C. (2013). Second-order optimality conditions in set-valued optimization via asymptotic derivatives. Optimization., 62, 743-758. https://doi.org/10.1080/02331934.2012.674948
Khan, A. A, Tammer, C., & Zlinescu, C. ( 2015). Set-Valued Optimization: An Introduction with Applications. Heidelberg: Springer.
Khanh, P. Q, and Tung, N. M. (2016). Second-order conditions for open-cone minimizers and firm minimizers in set-valued optimization subject to mixed constraints. J. Optim. Theory Appl., 171, 45-69. https://doi.org/10.1007/s10957-016-0995-x
Kuk, H., Tanino, T., & Tanaka, M. (1996a). Sensitivity analysis in parametrized convex vector optimization. J. Math. Anal. Appl., 202, 511-522. https://doi.org/10.1006/jmaa.1996.0331
Kuk, H., Tanino, T., & Tanaka, M. (1996b). Sensitivity analysis in vector optimization. J. Optim. Theory Appl., 89, 713-730. https://doi.org/10.1007/BF02275356
Li, S. J., Sun, X. K., & Zhai, J. (2012). Second-order contingent derivatives of set-valued mappings with application to set-valued optimization. Appl. Math. Comput., 218, 6874-6886. https://doi.org/10.1016/j.amc.2011.12.064
Li, S. J., Zhu, S. K., & Li, X. B. (2012). Second order optimality conditions for strict efficiency of constrained set-valued optimization. J. Optim. Theory Appl., 155, 534-557. https://doi.org/10.1007/s10957-012-0076-8
Li, S. J., & Zhai J. (2012). Second-order asymptotic differential properties and optimality conditions for weak vector variational inequalities. Optim. Lett., 6, 503-523. https://doi.org/10.1007/s11590-011-0276-4
Mordukhovich, B. S. (2006). Variational Analysis and Generalized Differentiation. Berlin: Springer, I: Basic Theory.
Penot, J. P. (1998). Second-order conditions for optimization problems with constraints. SIAM J. Control Optim., 37, 303-318. https://doi.org/10.1137/S0363012996311095
Sawaragi, Y., Nakayama, H., & Tanino, T. (1985). Theory of Multiobjective Optimization. New York: Academic Press.
Shi, D. S. (1991). Contingent derivative of the perturbation map in multiobjective optimization. J. Optim. Theory Appl., 70, 385-396. https://doi.org/10.1007/BF00940634
Shi, D. S. (1993). Sensitivity analysis in convex vector optimization. J. Optim. Theory Appl., 77, 145-159. https://doi.org/10.1007/BF00940783
Sun, X. K., & Li, S. J. (2014). Generalized second-order contingent epiderivatives in parametric vector optimization problem. J. Glob. Optim., 58, 351-363. https://doi.org/10.1007/s10898-013-0054-1
Tanino, T. (1988a). Sensitivity analysis in multiobjective optimization. J. Optim. Theory Appl., 56, 479-499. https://doi.org/10.1007/BF00939554
Tanino, T. (1988b). Stability and sensitivity analysis in convex vector optimization. SIAM J. Control Optim., 26, 521-536. https://doi.org/10.1137/0326031
Tung, L. T. (2017a). Second-order radial-asymptotic derivatives and applications in set-valued vector optimization. Pac. J. Optim., 13, 137-153. http://www.yokohamapublishers.jp/online-p/PJO/vol13/pjov13n1p137.pdf
Tung, L. T. (2017b). Variational sets and asymptotic variational sets of proper perturbation map in parametric vector optimization. Positivity., 21, 1647-1673. https://doi.org/10.1007/s11117-017-0491-z
Tung, L. T. (2018). On second-order proto-differentiability of perturbation maps. Set-Valued Var. Anal., 26, 561-579. https://doi.org/10.1007/s11228-016-0397-0
Tung, L. T. (2021). On higher-order proto-differentiability and higher-order asymptotic proto-differentiability of weak perturbation maps in parametric vector optimization. Positivity, 25(2), 579-604. https://doi.org/10.1007/s11117-020-00778-2
Tung, N. M. (2021). Second-order efficient optimality conditions for set-valued vector optimization in terms of asymptotic contingent epiderivatives. RAIRO Oper. Res., 55, 841-860. https://doi.org/10.1051/ro/2021039
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