On generalized second-order proto-differentiability of the benson proper perturbation maps in parametric vector optimization problems. (English) Zbl 1527.90231
Summary: This paper is concerned with second-order sensitivity analysis of parameterized vector optimization problems via generalized second-order contingent derivative. We prove that the Benson proper efficient solution map and the Benson proper efficient perturbation map of a parametric vector optimization problem are generalized second-order proto-differentiable under some suitable qualification conditions. Several examples are given to illustrate the obtained results.
MSC:
| 90C31 | Sensitivity, stability, parametric optimization |
| 90C29 | Multi-objective and goal programming |
| 90C26 | Nonconvex programming, global optimization |
Keywords:
parametric vector optimization problem; generalized second-order contingent derivative; Benson proper efficient solution map; Benson proper efficient perturbation map; sensitivity analysisReferences:
| [1] | Anh, NLH; Khanh, PQ, Variational sets of perturbation maps and applications to sensitivity analysis for constrained vector optimization, J. Optim. Theory Appl., 158, 363-384 (2013) · Zbl 1272.90076 |
| [2] | Anh, NLH, Sensitivity analysis in constrained set-valued optimization via Studniarski derivatives, Positivity, 21, 255-272 (2017) · Zbl 1369.49019 |
| [3] | Anh, NLH, Some results on sensitivity analysis in set-valued optimization, Positivity, 21, 1527-1543 (2017) · Zbl 1382.49048 |
| [4] | Anh, NLH, Second-order composed contingent derivatives of perturbation maps in set-valued optimization, Comput. Appl. Math., 38, 1-22 (2019) · Zbl 1438.46050 |
| [5] | Anh, NLH; Thoa, NT, Calculus rules of the generalized contingent derivative and applications to set-valued optimization, Positivity, 24, 81-94 (2020) · Zbl 1442.49019 |
| [6] | Aubin, JP; Frankowska, H., Set-Valued Analysis (1990), Boston: Birkhäuser, Boston · Zbl 0713.49021 |
| [7] | Bonnans, JF; Shapiro, A., Perturbation Analysis of Optimization Problems (2000), New York: Springer, New York · Zbl 0966.49001 |
| [8] | Chuong, TD; Yao, J-C, Coderivatives of efficient point multifunctions in parametric vector optimization, Taiwan J. Math., 13, 1671-1693 (2009) · Zbl 1192.49027 |
| [9] | Chuong, TD; Yao, J-C, Generalized clarke epiderivatives of parametric vector optimization problems, J. Optim. Theory Appl., 147, 77-94 (2010) · Zbl 1206.90158 |
| [10] | Chuong, TD, Clarke coderivatives of efficient point multifunctions in parametric vector optimization, Nonlinear Anal., 74, 273-285 (2011) · Zbl 1207.49031 |
| [11] | Chuong, TD, Derivatives of the efficient point multifunction in parametric vector optimization problems, J. Optim. Theory Appl., 156, 247-265 (2013) · Zbl 1282.90164 |
| [12] | Chuong, TD, Normal subdifferentials of effcient point multifunctions in parametric vector optimization, Optim. Lett., 7, 1087-1117 (2013) · Zbl 1282.90165 |
| [13] | Chuong, TD; Yao, J-C, Fréchet subdifferentials of efficient point multifunctions in parametric vector optimization, J. Glob. Optim., 57, 1229-1243 (2013) · Zbl 1277.49033 |
| [14] | Chuong, TD; Yao, J-C, Isolated calmness of efficient point multifunctions in parametric vector optimization, J. Nonlinear Convex Anal., 14, 719-734 (2013) · Zbl 1314.49014 |
| [15] | Diem, HTH; Khanh, PQ; Tung, LT, On higher-order sensitivity analysis in nonsmooth vector optimization, J. Optim. Theory Appl., 162, 463-488 (2014) · Zbl 1323.90061 |
| [16] | Huy, NQ; Lee, GM, On sensitivity analysis in vector optimization, Taiwan J. Math., 11, 945-958 (2007) · Zbl 1194.90086 |
| [17] | Huy, NQ; Lee, GM, Sensitivity of solutions to a parametric generalized equation, Set-Valued Anal., 16, 805-820 (2008) · Zbl 1156.90437 |
| [18] | Khan, A.; Tammer, C.; Zălinescu, C., Set-Valued Optimization: An Introduction with Applications (2015), Berlin: Springer, Berlin · Zbl 1308.49004 |
| [19] | Kuk, H.; Tanino, T.; Tanaka, M., Sensitivity analysis in parameterized convex vector optimization, J. Math. Anal. Appl., 202, 511-522 (1996) · Zbl 0856.90095 |
| [20] | Lee, GM; Huy, NQ, On proto-differentiability of generalized perturbation maps, J. Math. Anal. Appl., 324, 1297-1309 (2006) · Zbl 1104.49019 |
| [21] | Levy, AB; Rockafellar, RT, Sensitivity analysis of solutions to generalized equations, Trans. Am. Math. Soc., 345, 661-671 (1994) · Zbl 0815.47077 |
| [22] | Li, SJ; Liao, CM, Second-order differentiability of generalized perturbation maps, J. Glob. Optim., 52, 243-252 (2012) · Zbl 1266.90179 |
| [23] | Li, SJ; Sun, XK; Zhai, J., Second-order contingent derivatives of set-valued mappings with application to set-valued optimization, Appl. Math. Comput., 218, 6874-6886 (2012) · Zbl 1268.90096 |
| [24] | Luc, DT; Soleimani-damaneh, M.; Zamani, M., Semi-differentiability of the marginal mapping in vector optimization, SIAM J. Optim., 28, 1255-1281 (2018) · Zbl 1398.90155 |
| [25] | Mordukhovich, BS, Variational Analysis and Generalized Differentiation. I: Basic Theory (2006), Berlin: Springer, Berlin |
| [26] | Mordukhovich, BS, Variational Analysis and Generalized Differentiation. II: Applications (2006), Berlin: Springer, Berlin |
| [27] | Mordukhovich, BS, Variational Analysis and Applications (2018), New York: Springer, New York · Zbl 1402.49003 |
| [28] | Mordukhovich, BS; Outrata, JV, On second-order subdifferentials and their applications, SIAM J. Optim., 12, 139-169 (2001) · Zbl 1011.49016 |
| [29] | Mordukhovich, BS; Rockafellar, RT, Second-order subdifferential calculus with applications to tilt stability in optimization, SIAM J. Optim., 22, 953-986 (2012) · Zbl 1260.49022 |
| [30] | Mordukhovich, BS; Rockafellar, RT; Sarabi, ME, Characterizations of full stability in constrained optimization, SIAM J. Optim., 23, 1810-1849 (2013) · Zbl 1284.49032 |
| [31] | Mordukhovich, BS; Nghia, TTA; Rockafellar, RT, Full stability in finite-dimensional optimization, Math. Oper. Res., 40, 226-252 (2015) · Zbl 1308.90126 |
| [32] | Peng, Z.; Wan, Z., Second-order composed contingent derivative of the perturbation map in multiobjective optimization, Asia Pac. J. Oper. Res., 37, 2050002 (2020) · Zbl 1444.90107 |
| [33] | Rockafellar, RT, Proto-differentiablility of set-valued mappings and its applications in optimization, Ann. Inst. Non Linéaire. H. Poincaré Anal., 6, 449-482 (1989) · Zbl 0674.90082 |
| [34] | Rockafellar, RT; Wets, RJB, Variational Analysis (2009), Berlin: Springer, Berlin · Zbl 0888.49001 |
| [35] | Shi, DS, Contingent derivative of the perturbation map in multiobjective optimization, J. Optim. Theory Appl., 70, 385-396 (1991) · Zbl 0743.90092 |
| [36] | Shi, DS, Sensitivity analysis in convex vector optimization, J. Optim. Theory Appl., 77, 145-159 (1993) · Zbl 0796.90050 |
| [37] | Sun, XK; Li, SJ, Generalized second-order contingent epiderivatives in parametric vector optimization problem, J. Glob. Optim., 58, 351-363 (2014) · Zbl 1297.49027 |
| [38] | Taa, A., Set-valued derivatives of multifunctions and optimality conditions, Numer. Funct. Anal. Optim., 19, 121-140 (1998) · Zbl 1009.90106 |
| [39] | Tanino, T., Sensitivity analysis in multiobjective optimization, J. Optim. Theory Appl., 56, 479-499 (1988) · Zbl 0619.90073 |
| [40] | Tanino, T., Stability and sensitivity analysis in convex vector optimization, SIAM J. Control. Optim., 26, 521-536 (1988) · Zbl 0654.49011 |
| [41] | Tung, LT, Second-order radial-asymptotic derivatives and applications in set-valued vector optimization, Pac. J. Optim., 13, 137-153 (2017) · Zbl 1474.90425 |
| [42] | Tung, LT, Variational sets and asymptotic variational sets of proper perturbation map in parametric vector optimization, Positivity, 21, 1647-1673 (2017) · Zbl 1386.90155 |
| [43] | Tung, LT, On second-order proto-differentiability of perturbation maps, Set-Valued Var. Anal., 26, 561-579 (2018) · Zbl 1395.90238 |
| [44] | Tung, LT; Pham, TH, Sensitivity analysis in parametric vector optimization in Banach spaces via \(\tau^w-\) contingent derivatives, Turk. J. Math., 44, 152-168 (2020) · Zbl 1448.49048 |
| [45] | Tung, LT, On higher-order proto-differentiability of perturbation maps, Positivity, 24, 441-462 (2020) · Zbl 1453.90171 |
| [46] | Tung, LT, On higher-order proto-differentiability and higher-order asymptotic proto-differentiability of weak perturbation maps in parametric vector optimization, Positivity, 25, 579-604 (2021) · Zbl 1527.90232 |
| [47] | Tung, LT, On second-order composed proto-differentiability of proper perturbation maps in parametric vector optimization problems, Asia Pac. J. Oper. Res., 38, 2050040 (2021) · Zbl 1487.90596 |
| [48] | Wang, QL; Li, SJ; Teo, KL, Higher-order optimality conditions for weakly efficient solutions in nonconvex set-valued optimization, Optim. Lett., 4, 425-437 (2010) · Zbl 1229.90261 |
| [49] | Wang, QL; Li, SJ, Second-order contingent derivative of the perturbation map in multiobjective optimization, Fixed Point Theory Appl., 2011, 857520 (2011) · Zbl 1213.90224 |
| [50] | Wang, QL; Li, SJ, Sensitivity and stability for the second-order contingent derivative of the proper perturbation map in vector optimization, Optim. Lett., 6, 731-748 (2012) · Zbl 1280.90110 |
| [51] | Wang, QL; Zhang, XY, Second-order composed radial derivatives of the Benson proper perturbation map for parametric multi-objective optimization problems, Asia Pac. J. Oper. Res., 37, 2040011 (2020) · Zbl 1462.90124 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
