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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

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.
Citation: Qilin Wang, Xiao-Bing Li, Guolin Yu. Second-order weak composed epiderivatives and applications to optimality conditions. Journal of Industrial and 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.

[2]

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

[3]

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

[4]

J. Jahn and R. Rauh, Contingent epiderivatives and set-valued optimization, Math. Methods Oper. Res., 46 (1997), 193-211. 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), 187-200. doi: 10.1007/s001860050021.

[6]

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

[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.

[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.

[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.

[10]

J.-P. Penot, Second-order conditions for optimization problems with constraints, SIAM J. Control Optim., 37 (1999), 303-318. 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), 466-492. 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), 299-317. 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), 123-144. 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), 331-347. 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), 451-468. doi: 10.1007/BF02875738.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[24]

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

[25]

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

[26]

D. T. Luc, "Theory of Vector Optimization," Lecture Notes in Economics and Mathematical Systems, 319, Springer-Verlag, Berlin, 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), 365-375. 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 "Mathematical Analysis and Applications. Part A" (ed. L. Nachbin), Adv. in Math. Suppl. Stud., 7a, Academic Press, New York-London, (1981), 159-229.

[2]

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

[3]

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

[4]

J. Jahn and R. Rauh, Contingent epiderivatives and set-valued optimization, Math. Methods Oper. Res., 46 (1997), 193-211. 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), 187-200. doi: 10.1007/s001860050021.

[6]

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

[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.

[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.

[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.

[10]

J.-P. Penot, Second-order conditions for optimization problems with constraints, SIAM J. Control Optim., 37 (1999), 303-318. 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), 466-492. 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), 299-317. 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), 123-144. 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), 331-347. 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), 451-468. doi: 10.1007/BF02875738.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[24]

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

[25]

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

[26]

D. T. Luc, "Theory of Vector Optimization," Lecture Notes in Economics and Mathematical Systems, 319, Springer-Verlag, Berlin, 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), 365-375. doi: 10.1023/A:1021786303883.

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