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On linear vector optimization duality in infinite-dimensional spaces
Continuity of second-order adjacent derivatives for weak perturbation maps in vector optimization
1. | College of Mathematics and Science, Chongqing University, Chongqing 400044 |
2. | College of Mathematics and Statistics, Chongqing University, Chongqing, 401331 |
3. | Department of Mathematics and Statistics, Curtin University, G.P.O. Box U1987, Perth, WA 6845 |
References:
[1] |
A. V. Fiacco, "Introduction to Sensitivity and Stability Analysis in Nonlinear Programming," Academic Press, New York, 1983. |
[2] |
J. P. Aubin and H. Frankowska, "Set-Valued Analysis," Biekhäuser, Boston, 1990. |
[3] |
J. P. Aubin and I. Ekeland, "Applied Nonlinear Analysis," John Wiley, New York, 1984. |
[4] |
F. Ferro, An optimization result for set-valued mappings and a stability property in vector problems with constraints, J. Optim. Theory Appl., 90 (1996), 63-77.
doi: 10.1007/BF02192246. |
[5] |
R. B. Holmes, "Geometric Functional Analysis and Its Applications," Springer-Verlag, New York, 1975. |
[6] |
T. Tanino, Sensitivity analysis in multiobjective optimization, J. Optim. Theory Appl., 56 (1988), 479-499.
doi: 10.1007/BF00939554. |
[7] |
T. Tanino, Stability and sensitivity analysis in convex vector optimization, SIAM J. Control Optim., 26 (1988), 521-536.
doi: 10.1137/0326031. |
[8] |
H. Kuk, T. Tanino and M. Tanaka, Sensitivity analysis in parametrized convex vector optimization, J. Math. Anal. Appl., 202 (1996), 511-522.
doi: 10.1006/jmaa.1996.0331. |
[9] |
H. Kuk, T. Tanino and M. Tanaka, Sensitivity analysis in vector optimization, J. Optim. Theory Appl., 89 (1996), 713-730.
doi: 10.1007/BF02275356. |
[10] |
S. J. Li, Sensitivity and stability for contingent derivative in multiobjective optimization, Mathematica Applicata, 11 (1998), 49-53. |
[11] |
D. S. Shi, Contingent derivative of the perturbation map in multiobjective optimization, J. Optim. Theory Appl., 70 (1991), 385-396.
doi: 10.1007/BF00940634. |
[12] |
D. S. Shi, Sensitivity analysis in convex vector optimization, J. Optim. Theory Appl., 77 (1993), 145-159.
doi: 10.1007/BF00940783. |
[13] |
J. Jahn, A. A. Khan and P. Zeilinger, Second-order optimality conditions in set optimalization, J. Optim. Theory Appl., 125 (2005), 331-347.
doi: 10.1007/s10957-004-1841-0. |
[14] |
J. Jahn, "Vector Optimization-Theory, Applications and Extensions," Springer, 2004. |
[15] |
V. Kalashnikov, B.Jadamba and A.A. Khan, First and second-order optimality conditions in set optimization, in "Optimization with Multivalued Mappings"(eds. S. Dempe and V. Kalashnikov), Spring Science+Business Media, LLC, (2006), 265-276. |
[16] |
P. Q. Khanh and N. D. Tuan, Higher-order variational sets and higher-order optimality conditions for proper efficiency in set-valued nonsmooth vector optimization, J. Optim.Theory Appl., 139 (2008), 243-261.
doi: 10.1007/s10957-008-9414-2. |
[17] |
S. J. Li, K. L. Teo and X. Q. Yang, Higher-order optimality conditions for set-valued optimization, J. Optim. Theory Appl., 137 (2008), 533-553.
doi: 10.1007/s10957-007-9345-3. |
[18] |
S. J. Li and C. R. Chen, Higher-order optimality conditions for Henig efficient solutions in set-valued optimization, J. Math. Anal. Appl., 323 (2006), 1184-1200.
doi: 10.1016/j.jmaa.2005.11.035. |
[19] |
Q. L. Wang and S. J. Li, Generalized higher-order optimality conditions for set-valued optimization under Henig efficiency, Numer. Funct. Anal. Optim., 30 (2009), 849-869.
doi: 10.1080/01630560903139540. |
[20] |
Q. L. Wang and S. J. Li, Higher-Order Weakly Generalized Adjacent Epiderivatives and Applications to Duality of Set-Valued Optimization, J. Inequal. Appl., 2009, 18 pages. |
[21] |
Q. L. Wang, S. J. Li and K. L. Teo, Higher-order optimality conditions for weakly efficient solutions in nonconvex set-valued optimization, Optim. Lett. , 4 (2010), 425-437.
doi: 10.1007/s11590-009-0170-5. |
[22] |
D. T. Luc, "Theory of Vector Optimization," Springer, Berlin, 1989. |
[23] |
S. W. Xiang and W. S. Yin, Stability results for efficient solutions of vector optimization problems, J. Optim. Theory Appl., 134 (2007), 385-398.
doi: 10.1007/s10957-007-9214-0. |
[24] |
J. F. Bonnans and A. Shapiro, "Perturbation Analysis of Optimization Problems," Springer-Verlag, New York, 2000. |
show all references
References:
[1] |
A. V. Fiacco, "Introduction to Sensitivity and Stability Analysis in Nonlinear Programming," Academic Press, New York, 1983. |
[2] |
J. P. Aubin and H. Frankowska, "Set-Valued Analysis," Biekhäuser, Boston, 1990. |
[3] |
J. P. Aubin and I. Ekeland, "Applied Nonlinear Analysis," John Wiley, New York, 1984. |
[4] |
F. Ferro, An optimization result for set-valued mappings and a stability property in vector problems with constraints, J. Optim. Theory Appl., 90 (1996), 63-77.
doi: 10.1007/BF02192246. |
[5] |
R. B. Holmes, "Geometric Functional Analysis and Its Applications," Springer-Verlag, New York, 1975. |
[6] |
T. Tanino, Sensitivity analysis in multiobjective optimization, J. Optim. Theory Appl., 56 (1988), 479-499.
doi: 10.1007/BF00939554. |
[7] |
T. Tanino, Stability and sensitivity analysis in convex vector optimization, SIAM J. Control Optim., 26 (1988), 521-536.
doi: 10.1137/0326031. |
[8] |
H. Kuk, T. Tanino and M. Tanaka, Sensitivity analysis in parametrized convex vector optimization, J. Math. Anal. Appl., 202 (1996), 511-522.
doi: 10.1006/jmaa.1996.0331. |
[9] |
H. Kuk, T. Tanino and M. Tanaka, Sensitivity analysis in vector optimization, J. Optim. Theory Appl., 89 (1996), 713-730.
doi: 10.1007/BF02275356. |
[10] |
S. J. Li, Sensitivity and stability for contingent derivative in multiobjective optimization, Mathematica Applicata, 11 (1998), 49-53. |
[11] |
D. S. Shi, Contingent derivative of the perturbation map in multiobjective optimization, J. Optim. Theory Appl., 70 (1991), 385-396.
doi: 10.1007/BF00940634. |
[12] |
D. S. Shi, Sensitivity analysis in convex vector optimization, J. Optim. Theory Appl., 77 (1993), 145-159.
doi: 10.1007/BF00940783. |
[13] |
J. Jahn, A. A. Khan and P. Zeilinger, Second-order optimality conditions in set optimalization, J. Optim. Theory Appl., 125 (2005), 331-347.
doi: 10.1007/s10957-004-1841-0. |
[14] |
J. Jahn, "Vector Optimization-Theory, Applications and Extensions," Springer, 2004. |
[15] |
V. Kalashnikov, B.Jadamba and A.A. Khan, First and second-order optimality conditions in set optimization, in "Optimization with Multivalued Mappings"(eds. S. Dempe and V. Kalashnikov), Spring Science+Business Media, LLC, (2006), 265-276. |
[16] |
P. Q. Khanh and N. D. Tuan, Higher-order variational sets and higher-order optimality conditions for proper efficiency in set-valued nonsmooth vector optimization, J. Optim.Theory Appl., 139 (2008), 243-261.
doi: 10.1007/s10957-008-9414-2. |
[17] |
S. J. Li, K. L. Teo and X. Q. Yang, Higher-order optimality conditions for set-valued optimization, J. Optim. Theory Appl., 137 (2008), 533-553.
doi: 10.1007/s10957-007-9345-3. |
[18] |
S. J. Li and C. R. Chen, Higher-order optimality conditions for Henig efficient solutions in set-valued optimization, J. Math. Anal. Appl., 323 (2006), 1184-1200.
doi: 10.1016/j.jmaa.2005.11.035. |
[19] |
Q. L. Wang and S. J. Li, Generalized higher-order optimality conditions for set-valued optimization under Henig efficiency, Numer. Funct. Anal. Optim., 30 (2009), 849-869.
doi: 10.1080/01630560903139540. |
[20] |
Q. L. Wang and S. J. Li, Higher-Order Weakly Generalized Adjacent Epiderivatives and Applications to Duality of Set-Valued Optimization, J. Inequal. Appl., 2009, 18 pages. |
[21] |
Q. L. Wang, S. J. Li and K. L. Teo, Higher-order optimality conditions for weakly efficient solutions in nonconvex set-valued optimization, Optim. Lett. , 4 (2010), 425-437.
doi: 10.1007/s11590-009-0170-5. |
[22] |
D. T. Luc, "Theory of Vector Optimization," Springer, Berlin, 1989. |
[23] |
S. W. Xiang and W. S. Yin, Stability results for efficient solutions of vector optimization problems, J. Optim. Theory Appl., 134 (2007), 385-398.
doi: 10.1007/s10957-007-9214-0. |
[24] |
J. F. Bonnans and A. Shapiro, "Perturbation Analysis of Optimization Problems," Springer-Verlag, New York, 2000. |
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