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April  2013, 9(2): 455-470. doi: 10.3934/jimo.2013.9.455

Second-order weak composed epiderivatives and applications to optimality conditions

Qilin Wang 1, , Xiao-Bing Li 1, and  Guolin Yu 2,

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

Received  April 2011 Revised  January 2013 Published  February 2013

In this paper, one introduces the second-order weak composed contingent epiderivative of set-valued maps, and discusses some of its properties. Then, by virtue of the second-order weak composed contingent epiderivative, necessary optimality conditions and sufficient optimality conditions are obtained for set-valued optimization problems. As consequences, recent existing results are derived. Several examples are provided to show the main results obtained.
Keywords: Set-valued optimization, second-order optimality conditions., second-order weak composed contingent epiderivatives.
Mathematics Subject Classification: 90C26, 90C4.
Citation: Qilin Wang, Xiao-Bing Li, Guolin Yu. Second-order weak composed epiderivatives and applications to optimality conditions. Journal of Industrial & Management Optimization, 2013, 9 (2) : 455-470. doi: 10.3934/jimo.2013.9.455
References:
[1]

J.-P. Aubin, Contingent derivatives of set-valued maps and existence of solutions to nonlinear inclusions and differential inclusions, in "Mathematical Analysis and Applications. Part A" (ed. L. Nachbin), Adv. in Math. Suppl. Stud., 7a, Academic Press, New York-London, (1981), 159-229.  Google Scholar

[2]

J.-P. Aubin and H. Frankowska, "Set-Valued Analysis," Systems & Control: Foundations & Applications, 2, Birkhäuser Boston, Inc., Boston, 1990.  Google Scholar

[3]

H. W. Corley, Optimality condition for maximizations of set-valued functions, J. Optim. Theory Appl., 58 (1988), 1-10. doi: 10.1007/BF00939767.  Google Scholar

[4]

J. Jahn and R. Rauh, Contingent epiderivatives and set-valued optimization, Math. Methods Oper. Res., 46 (1997), 193-211. doi: 10.1007/BF01217690.  Google Scholar

[5]

G. Y. Chen and J. Jahn, Optimality conditions for set-valued optimization problems. Set-valued optimization, Math. Methods Oper. Res., 48 (1998), 187-200. doi: 10.1007/s001860050021.  Google Scholar

[6]

E. M. Bednarczuk and W. Song, Contingent epiderivative and its applications to set-valued optimization, Control Cybernet., 27 (1998), 375-386.  Google Scholar

[7]

G. Bigi and M. Castellani, K-epiderivatives for set-valued functions and optimization, Math. Methods Oper. Res., 55 (2002), 401-412. doi: 10.1007/s001860200187.  Google Scholar

[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), 332-350. doi: 10.1016/S0022-247X(03)00360-3.  Google Scholar

[9]

J. Jahn and A. A. Khan, Generalized contingent epiderivatives in set-valued optimization: Optimality conditions, Numer. Funct. Anal. Optim., 23 (2002), 807-831. doi: 10.1081/NFA-120016271.  Google Scholar

[10]

J.-P. Penot, Second-order conditions for optimization problems with constraints, SIAM J. Control Optim., 37 (1999), 303-318. doi: 10.1137/S0363012996311095.  Google Scholar

[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), 466-492. doi: 10.1137/S1052623496306760.  Google Scholar

[12]

B. Jiménez and V. Novo, Second-order necessary conditions in set constrained differentiable vector optimization, Math. Methods Oper. Res., 58 (2003), 299-317. doi: 10.1007/s001860300283.  Google Scholar

[13]

B. Jiménez and V. Novo, Optimality conditions in differentiable vector optimization via second-order tangent sets, Appl. Math. Optim., 49 (2004), 123-144. doi: 10.1007/s00245-003-0782-6.  Google Scholar

[14]

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.  Google Scholar

[15]

M. Durea, First and second order optimality conditions for set-valued optimization problems, Rend. Circ. Mat. Palermo (2), 53 (2004), 451-468. doi: 10.1007/BF02875738.  Google Scholar

[16]

G. Bigi, On sufficient second order optimality conditions in multiobjective optimization, Math. Methods Oper. Res., 63 (2006), 77-85. doi: 10.1007/s00186-005-0013-9.  Google Scholar

[17]

L. Rodríguez-Marín and M. Sama, About contingent epiderivatives, J. Math. Anal. Appl., 327 (2007), 745-762. doi: 10.1016/j.jmaa.2006.04.060.  Google Scholar

[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), 587-604. doi: 10.1007/s10957-011-9915-2.  Google Scholar

[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), 1184-1200. doi: 10.1016/j.jmaa.2005.11.035.  Google Scholar

[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), 533-553. doi: 10.1007/s10957-007-9345-3.  Google Scholar

[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), 243-261. doi: 10.1007/s10957-008-9414-2.  Google Scholar

[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), 1389-1399. doi: 10.1016/j.camwa.2009.01.012.  Google Scholar

[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), 425-437. doi: 10.1007/s11590-009-0170-5.  Google Scholar

[24]

J. F. Bonnans and A. Shapiro, "Perturbation Analysis of Optimization Problems," Springer Series in Operations Research, Springer-Verlag, New York, 2000.  Google Scholar

[25]

J. Jahn, "Vector Optimization. Theory, Applications, and Extensions," Springer-Verlag, Berlin, 2004.  Google Scholar

[26]

D. T. Luc, "Theory of Vector Optimization," Lecture Notes in Economics and Mathematical Systems, 319, Springer-Verlag, Berlin, 1989.  Google Scholar

[27]

Z. Li, A theorem of the alternative and its application to the optimization of set-valued maps, J. Optim. Theory Appl., 100 (1999), 365-375. doi: 10.1023/A:1021786303883.  Google Scholar

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 "Mathematical Analysis and Applications. Part A" (ed. L. Nachbin), Adv. in Math. Suppl. Stud., 7a, Academic Press, New York-London, (1981), 159-229.  Google Scholar

[2]

J.-P. Aubin and H. Frankowska, "Set-Valued Analysis," Systems & Control: Foundations & Applications, 2, Birkhäuser Boston, Inc., Boston, 1990.  Google Scholar

[3]

H. W. Corley, Optimality condition for maximizations of set-valued functions, J. Optim. Theory Appl., 58 (1988), 1-10. doi: 10.1007/BF00939767.  Google Scholar

[4]

J. Jahn and R. Rauh, Contingent epiderivatives and set-valued optimization, Math. Methods Oper. Res., 46 (1997), 193-211. doi: 10.1007/BF01217690.  Google Scholar

[5]

G. Y. Chen and J. Jahn, Optimality conditions for set-valued optimization problems. Set-valued optimization, Math. Methods Oper. Res., 48 (1998), 187-200. doi: 10.1007/s001860050021.  Google Scholar

[6]

E. M. Bednarczuk and W. Song, Contingent epiderivative and its applications to set-valued optimization, Control Cybernet., 27 (1998), 375-386.  Google Scholar

[7]

G. Bigi and M. Castellani, K-epiderivatives for set-valued functions and optimization, Math. Methods Oper. Res., 55 (2002), 401-412. doi: 10.1007/s001860200187.  Google Scholar

[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), 332-350. doi: 10.1016/S0022-247X(03)00360-3.  Google Scholar

[9]

J. Jahn and A. A. Khan, Generalized contingent epiderivatives in set-valued optimization: Optimality conditions, Numer. Funct. Anal. Optim., 23 (2002), 807-831. doi: 10.1081/NFA-120016271.  Google Scholar

[10]

J.-P. Penot, Second-order conditions for optimization problems with constraints, SIAM J. Control Optim., 37 (1999), 303-318. doi: 10.1137/S0363012996311095.  Google Scholar

[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), 466-492. doi: 10.1137/S1052623496306760.  Google Scholar

[12]

B. Jiménez and V. Novo, Second-order necessary conditions in set constrained differentiable vector optimization, Math. Methods Oper. Res., 58 (2003), 299-317. doi: 10.1007/s001860300283.  Google Scholar

[13]

B. Jiménez and V. Novo, Optimality conditions in differentiable vector optimization via second-order tangent sets, Appl. Math. Optim., 49 (2004), 123-144. doi: 10.1007/s00245-003-0782-6.  Google Scholar

[14]

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.  Google Scholar

[15]

M. Durea, First and second order optimality conditions for set-valued optimization problems, Rend. Circ. Mat. Palermo (2), 53 (2004), 451-468. doi: 10.1007/BF02875738.  Google Scholar

[16]

G. Bigi, On sufficient second order optimality conditions in multiobjective optimization, Math. Methods Oper. Res., 63 (2006), 77-85. doi: 10.1007/s00186-005-0013-9.  Google Scholar

[17]

L. Rodríguez-Marín and M. Sama, About contingent epiderivatives, J. Math. Anal. Appl., 327 (2007), 745-762. doi: 10.1016/j.jmaa.2006.04.060.  Google Scholar

[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), 587-604. doi: 10.1007/s10957-011-9915-2.  Google Scholar

[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), 1184-1200. doi: 10.1016/j.jmaa.2005.11.035.  Google Scholar

[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), 533-553. doi: 10.1007/s10957-007-9345-3.  Google Scholar

[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), 243-261. doi: 10.1007/s10957-008-9414-2.  Google Scholar

[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), 1389-1399. doi: 10.1016/j.camwa.2009.01.012.  Google Scholar

[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), 425-437. doi: 10.1007/s11590-009-0170-5.  Google Scholar

[24]

J. F. Bonnans and A. Shapiro, "Perturbation Analysis of Optimization Problems," Springer Series in Operations Research, Springer-Verlag, New York, 2000.  Google Scholar

[25]

J. Jahn, "Vector Optimization. Theory, Applications, and Extensions," Springer-Verlag, Berlin, 2004.  Google Scholar

[26]

D. T. Luc, "Theory of Vector Optimization," Lecture Notes in Economics and Mathematical Systems, 319, Springer-Verlag, Berlin, 1989.  Google Scholar

[27]

Z. Li, A theorem of the alternative and its application to the optimization of set-valued maps, J. Optim. Theory Appl., 100 (1999), 365-375. doi: 10.1023/A:1021786303883.  Google Scholar

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