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Second-order weak composed epiderivatives and applications to optimality conditions
1. | College of Sciences, Chongqing Jiaotong University, Chongqing, 400074, China, China |
2. | Research Institute of Information and System Computation Science, Beifang University of Nationalities, Yinchuan, 750021, China |
References:
[1] |
J.-P. Aubin, Contingent derivatives of set-valued maps and existence of solutions to nonlinear inclusions and differential inclusions,, in, 7a (1981), 159.
|
[2] |
J.-P. Aubin and H. Frankowska, "Set-Valued Analysis,", Systems & Control: Foundations & Applications, 2 (1990).
|
[3] |
H. W. Corley, Optimality condition for maximizations of set-valued functions,, J. Optim. Theory Appl., 58 (1988), 1.
doi: 10.1007/BF00939767. |
[4] |
J. Jahn and R. Rauh, Contingent epiderivatives and set-valued optimization,, Math. Methods Oper. Res., 46 (1997), 193.
doi: 10.1007/BF01217690. |
[5] |
G. Y. Chen and J. Jahn, Optimality conditions for set-valued optimization problems. Set-valued optimization,, Math. Methods Oper. Res., 48 (1998), 187.
doi: 10.1007/s001860050021. |
[6] |
E. M. Bednarczuk and W. Song, Contingent epiderivative and its applications to set-valued optimization,, Control Cybernet., 27 (1998), 375.
|
[7] |
G. Bigi and M. Castellani, K-epiderivatives for set-valued functions and optimization,, Math. Methods Oper. Res., 55 (2002), 401.
doi: 10.1007/s001860200187. |
[8] |
X.-H. Gong, H.-B. Dong and S.-Y. Wang, Optimality conditions for proper efficient solutions of vector set-valued optimization,, J. Math. Anal. Appl., 284 (2003), 332.
doi: 10.1016/S0022-247X(03)00360-3. |
[9] |
J. Jahn and A. A. Khan, Generalized contingent epiderivatives in set-valued optimization: Optimality conditions,, Numer. Funct. Anal. Optim., 23 (2002), 807.
doi: 10.1081/NFA-120016271. |
[10] |
J.-P. Penot, Second-order conditions for optimization problems with constraints,, SIAM J. Control Optim., 37 (1999), 303.
doi: 10.1137/S0363012996311095. |
[11] |
J. F. Bonnans, R. Cominetti and A. Shapiro, Second-order optimality conditions based on parabolic second-order tangent sets,, SIAM J. Optim., 9 (1999), 466.
doi: 10.1137/S1052623496306760. |
[12] |
B. Jiménez and V. Novo, Second-order necessary conditions in set constrained differentiable vector optimization,, Math. Methods Oper. Res., 58 (2003), 299.
doi: 10.1007/s001860300283. |
[13] |
B. Jiménez and V. Novo, Optimality conditions in differentiable vector optimization via second-order tangent sets,, Appl. Math. Optim., 49 (2004), 123.
doi: 10.1007/s00245-003-0782-6. |
[14] |
J. Jahn, A. A. Khan and P. Zeilinger, Second-order optimality conditions in set optimalization,, J. Optim. Theory Appl., 125 (2005), 331.
doi: 10.1007/s10957-004-1841-0. |
[15] |
M. Durea, First and second order optimality conditions for set-valued optimization problems,, Rend. Circ. Mat. Palermo (2), 53 (2004), 451.
doi: 10.1007/BF02875738. |
[16] |
G. Bigi, On sufficient second order optimality conditions in multiobjective optimization,, Math. Methods Oper. Res., 63 (2006), 77.
doi: 10.1007/s00186-005-0013-9. |
[17] |
L. Rodríguez-Marín and M. Sama, About contingent epiderivatives,, J. Math. Anal. Appl., 327 (2007), 745.
doi: 10.1016/j.jmaa.2006.04.060. |
[18] |
S. J. Li, S. K. Zhu and K. L. Teo, New generalized second-order contingent epiderivatives and set-valued optimization problems,, J. Optim. Theory Appl., 152 (2012), 587.
doi: 10.1007/s10957-011-9915-2. |
[19] |
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.
doi: 10.1016/j.jmaa.2005.11.035. |
[20] |
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.
doi: 10.1007/s10957-007-9345-3. |
[21] |
P. Q. Khanh and N. D. Tuan, Higher-order variational sets and higher-order optimality conditions for proper efficiency in set-valued nonsmooth vector optimalization,, J. Optim. Theory Appl., 139 (2008), 243.
doi: 10.1007/s10957-008-9414-2. |
[22] |
C. R. Chen, S. J. Li and K. L. Teo, Higher order weak epiderivatives and applications to duality and optimality conditions,, Comput. Math. Appl., 57 (2009), 1389.
doi: 10.1016/j.camwa.2009.01.012. |
[23] |
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.
doi: 10.1007/s11590-009-0170-5. |
[24] |
J. F. Bonnans and A. Shapiro, "Perturbation Analysis of Optimization Problems,", Springer Series in Operations Research, (2000).
|
[25] |
J. Jahn, "Vector Optimization. Theory, Applications, and Extensions,", Springer-Verlag, (2004).
|
[26] |
D. T. Luc, "Theory of Vector Optimization,", Lecture Notes in Economics and Mathematical Systems, 319 (1989).
|
[27] |
Z. Li, A theorem of the alternative and its application to the optimization of set-valued maps,, J. Optim. Theory Appl., 100 (1999), 365.
doi: 10.1023/A:1021786303883. |
show all references
References:
[1] |
J.-P. Aubin, Contingent derivatives of set-valued maps and existence of solutions to nonlinear inclusions and differential inclusions,, in, 7a (1981), 159.
|
[2] |
J.-P. Aubin and H. Frankowska, "Set-Valued Analysis,", Systems & Control: Foundations & Applications, 2 (1990).
|
[3] |
H. W. Corley, Optimality condition for maximizations of set-valued functions,, J. Optim. Theory Appl., 58 (1988), 1.
doi: 10.1007/BF00939767. |
[4] |
J. Jahn and R. Rauh, Contingent epiderivatives and set-valued optimization,, Math. Methods Oper. Res., 46 (1997), 193.
doi: 10.1007/BF01217690. |
[5] |
G. Y. Chen and J. Jahn, Optimality conditions for set-valued optimization problems. Set-valued optimization,, Math. Methods Oper. Res., 48 (1998), 187.
doi: 10.1007/s001860050021. |
[6] |
E. M. Bednarczuk and W. Song, Contingent epiderivative and its applications to set-valued optimization,, Control Cybernet., 27 (1998), 375.
|
[7] |
G. Bigi and M. Castellani, K-epiderivatives for set-valued functions and optimization,, Math. Methods Oper. Res., 55 (2002), 401.
doi: 10.1007/s001860200187. |
[8] |
X.-H. Gong, H.-B. Dong and S.-Y. Wang, Optimality conditions for proper efficient solutions of vector set-valued optimization,, J. Math. Anal. Appl., 284 (2003), 332.
doi: 10.1016/S0022-247X(03)00360-3. |
[9] |
J. Jahn and A. A. Khan, Generalized contingent epiderivatives in set-valued optimization: Optimality conditions,, Numer. Funct. Anal. Optim., 23 (2002), 807.
doi: 10.1081/NFA-120016271. |
[10] |
J.-P. Penot, Second-order conditions for optimization problems with constraints,, SIAM J. Control Optim., 37 (1999), 303.
doi: 10.1137/S0363012996311095. |
[11] |
J. F. Bonnans, R. Cominetti and A. Shapiro, Second-order optimality conditions based on parabolic second-order tangent sets,, SIAM J. Optim., 9 (1999), 466.
doi: 10.1137/S1052623496306760. |
[12] |
B. Jiménez and V. Novo, Second-order necessary conditions in set constrained differentiable vector optimization,, Math. Methods Oper. Res., 58 (2003), 299.
doi: 10.1007/s001860300283. |
[13] |
B. Jiménez and V. Novo, Optimality conditions in differentiable vector optimization via second-order tangent sets,, Appl. Math. Optim., 49 (2004), 123.
doi: 10.1007/s00245-003-0782-6. |
[14] |
J. Jahn, A. A. Khan and P. Zeilinger, Second-order optimality conditions in set optimalization,, J. Optim. Theory Appl., 125 (2005), 331.
doi: 10.1007/s10957-004-1841-0. |
[15] |
M. Durea, First and second order optimality conditions for set-valued optimization problems,, Rend. Circ. Mat. Palermo (2), 53 (2004), 451.
doi: 10.1007/BF02875738. |
[16] |
G. Bigi, On sufficient second order optimality conditions in multiobjective optimization,, Math. Methods Oper. Res., 63 (2006), 77.
doi: 10.1007/s00186-005-0013-9. |
[17] |
L. Rodríguez-Marín and M. Sama, About contingent epiderivatives,, J. Math. Anal. Appl., 327 (2007), 745.
doi: 10.1016/j.jmaa.2006.04.060. |
[18] |
S. J. Li, S. K. Zhu and K. L. Teo, New generalized second-order contingent epiderivatives and set-valued optimization problems,, J. Optim. Theory Appl., 152 (2012), 587.
doi: 10.1007/s10957-011-9915-2. |
[19] |
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.
doi: 10.1016/j.jmaa.2005.11.035. |
[20] |
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.
doi: 10.1007/s10957-007-9345-3. |
[21] |
P. Q. Khanh and N. D. Tuan, Higher-order variational sets and higher-order optimality conditions for proper efficiency in set-valued nonsmooth vector optimalization,, J. Optim. Theory Appl., 139 (2008), 243.
doi: 10.1007/s10957-008-9414-2. |
[22] |
C. R. Chen, S. J. Li and K. L. Teo, Higher order weak epiderivatives and applications to duality and optimality conditions,, Comput. Math. Appl., 57 (2009), 1389.
doi: 10.1016/j.camwa.2009.01.012. |
[23] |
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.
doi: 10.1007/s11590-009-0170-5. |
[24] |
J. F. Bonnans and A. Shapiro, "Perturbation Analysis of Optimization Problems,", Springer Series in Operations Research, (2000).
|
[25] |
J. Jahn, "Vector Optimization. Theory, Applications, and Extensions,", Springer-Verlag, (2004).
|
[26] |
D. T. Luc, "Theory of Vector Optimization,", Lecture Notes in Economics and Mathematical Systems, 319 (1989).
|
[27] |
Z. Li, A theorem of the alternative and its application to the optimization of set-valued maps,, J. Optim. Theory Appl., 100 (1999), 365.
doi: 10.1023/A:1021786303883. |
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