2013, 3(4): 643-653. doi: 10.3934/naco.2013.3.643

Characterizations of the $E$-Benson proper efficiency in vector optimization problems

1. 

College of Mathematics Science, Chongqing Normal University, Chongqing 401331, China, China

Received  August 2013 Revised  October 2013 Published  October 2013

In this paper, under the nearly $E$-subconvexlikeness, some characterizations of the $E$-Benson proper efficiency are established in terms of scalarization, Lagrange multipliers, saddle point criteria and duality for a vector optimization problem with set-valued maps. Our main results generalize and unify some previously known results.
Citation: Kequan Zhao, Xinmin Yang. Characterizations of the $E$-Benson proper efficiency in vector optimization problems. Numerical Algebra, Control and Optimization, 2013, 3 (4) : 643-653. doi: 10.3934/naco.2013.3.643
References:
[1]

H. P. Benson, An improved definition of proper efficiency for vector maximization with respect to cones, J. Math. Anal. Appl., 71 (1979), 232-241. doi: 10.1016/0022-247X(79)90226-9.

[2]

J. Borwein, Proper efficient points for maximizations with respect to cones, SIAM. J. Control and Optim., 15 (1977), 57-63.

[3]

G. Y. Chen and W. D. Rong, Characterizations of the Benson proper efficiency for nonconvex vector optimization, J. Optim. Theory Appl., 98 (1998), 365-384. doi: 10.1023/A:1022689517921.

[4]

G. Y. Chen, X. X. Huang and X. Q. Yang, "Vector Optimization. Lecture Notes in Economics and Mathematical Sciences, 541," Springer, Berlin, 2005.

[5]

M. Chicco, F. Mignanego, L. Pusillo and S. Tijs, Vector optimization problems via improvement sets, J. Optim. Theory Appl., 150 (2011), 516-529. doi: 10.1007/s10957-011-9851-1.

[6]

M. Ehrgott, "Multicriteria Optimization," Springer, Berlin, 2005.

[7]

Y. Gao and X. M. Yang, Optimality conditions for approximate solutions of vector optimization problems, J. Ind. Manag. Optim., 7 (2011), 483-496. doi: 10.3934/jimo.2011.7.483.

[8]

A. M. Geffrion, Proper efficiency and the theory of vector maximization, J. Math. Anal. Appl., 22 (1968), 618-630.

[9]

B. A. Ghaznavi-ghosoni, E. Khorram and M. Soleimani-damaneh, Scalarization for characterization of approximate strong/weak/proper efficiency in multiobjective optimization, Optimization, 62 (2013), 703-720. doi: 10.1080/02331934.2012.668190.

[10]

C. Gutiérrez, B. Jiménez and V. Novo, Improvement sets and vector optimization, Eur. J. Oper. Res., 223 (2012), 304-311.

[11]

C. Gutiérrez, B. Jiménez and V. Novo, A unified approach and optimality conditions for approximate solutions of vector optimization problems, SIAM J. Optim., 17 (2006), 688-710.

[12]

C. Gutiérrez, L. Huerga and V. Novo, Scalarization and saddle points of approximate proper solutions in nearly subconvexlike vector optimization problems, J. Math. Anal. Appl., 389 (2012), 1046-1058. doi: 10.1016/j.jmaa.2011.12.050.

[13]

M. I. Henig, Proper efficiency with respect to cones, J. Optim. Theory Appl., 36 (1982), 387-407. doi: 10.1007/BF00934353.

[14]

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

[15]

Z. F. Li, Benson proper efficiency in the vector optimization of set-valued maps, J. Optim. Theory Appl., 98 (1998), 623-649. doi: 10.1023/A:1022676013609.

[16]

J. C. Liu, ε-Properly efficient solutions to nondifferentiable multiobjective programming problems, Appl. Math. Lett., 12 (1999), 109-113. doi: 10.1016/S0893-9659(99)00087-7.

[17]

D. T. Luc, "Theory of Vector Optimization. Lecture Notes in Economics and Mathematical Sciences, 319," Springer, Berlin, 1988.

[18]

W. D. Rong and Y. Ma, ε-Properly efficient solutions of vector optimization problems with set-valued maps, OR Transactions, 4 (2000), 21-32.

[19]

X. M. Yang, D. Li and S. Y. Wang, Near-subconvexlikeness in vector optimization with set-valued functions, J. Optim. Theory Appl., 110 (2001), 413-427. doi: 10.1023/A:1017535631418.

[20]

X. M. Yang, X. Q. Yang and G. Y. Chen, Theorems of the alternative and optimization with set-valued maps, J. Optim. Theory Appl., 107 (2000), 627-640. doi: 10.1023/A:1004613630675.

[21]

K. Q. Zhao and X. M. Yang, E-Benson proper efficiency in vector optimization, Optimization, doi:10.1080/02331934.2013.798321, 2013. doi: 10.1080/02331934.2013.798321.

[22]

K. Q. Zhao, X. M. Yang and J. W. Peng, Weak E-Optimal solution in vector optimization, Taiwan. J. Math., 17 (2013), 1287-1302.

show all references

References:
[1]

H. P. Benson, An improved definition of proper efficiency for vector maximization with respect to cones, J. Math. Anal. Appl., 71 (1979), 232-241. doi: 10.1016/0022-247X(79)90226-9.

[2]

J. Borwein, Proper efficient points for maximizations with respect to cones, SIAM. J. Control and Optim., 15 (1977), 57-63.

[3]

G. Y. Chen and W. D. Rong, Characterizations of the Benson proper efficiency for nonconvex vector optimization, J. Optim. Theory Appl., 98 (1998), 365-384. doi: 10.1023/A:1022689517921.

[4]

G. Y. Chen, X. X. Huang and X. Q. Yang, "Vector Optimization. Lecture Notes in Economics and Mathematical Sciences, 541," Springer, Berlin, 2005.

[5]

M. Chicco, F. Mignanego, L. Pusillo and S. Tijs, Vector optimization problems via improvement sets, J. Optim. Theory Appl., 150 (2011), 516-529. doi: 10.1007/s10957-011-9851-1.

[6]

M. Ehrgott, "Multicriteria Optimization," Springer, Berlin, 2005.

[7]

Y. Gao and X. M. Yang, Optimality conditions for approximate solutions of vector optimization problems, J. Ind. Manag. Optim., 7 (2011), 483-496. doi: 10.3934/jimo.2011.7.483.

[8]

A. M. Geffrion, Proper efficiency and the theory of vector maximization, J. Math. Anal. Appl., 22 (1968), 618-630.

[9]

B. A. Ghaznavi-ghosoni, E. Khorram and M. Soleimani-damaneh, Scalarization for characterization of approximate strong/weak/proper efficiency in multiobjective optimization, Optimization, 62 (2013), 703-720. doi: 10.1080/02331934.2012.668190.

[10]

C. Gutiérrez, B. Jiménez and V. Novo, Improvement sets and vector optimization, Eur. J. Oper. Res., 223 (2012), 304-311.

[11]

C. Gutiérrez, B. Jiménez and V. Novo, A unified approach and optimality conditions for approximate solutions of vector optimization problems, SIAM J. Optim., 17 (2006), 688-710.

[12]

C. Gutiérrez, L. Huerga and V. Novo, Scalarization and saddle points of approximate proper solutions in nearly subconvexlike vector optimization problems, J. Math. Anal. Appl., 389 (2012), 1046-1058. doi: 10.1016/j.jmaa.2011.12.050.

[13]

M. I. Henig, Proper efficiency with respect to cones, J. Optim. Theory Appl., 36 (1982), 387-407. doi: 10.1007/BF00934353.

[14]

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

[15]

Z. F. Li, Benson proper efficiency in the vector optimization of set-valued maps, J. Optim. Theory Appl., 98 (1998), 623-649. doi: 10.1023/A:1022676013609.

[16]

J. C. Liu, ε-Properly efficient solutions to nondifferentiable multiobjective programming problems, Appl. Math. Lett., 12 (1999), 109-113. doi: 10.1016/S0893-9659(99)00087-7.

[17]

D. T. Luc, "Theory of Vector Optimization. Lecture Notes in Economics and Mathematical Sciences, 319," Springer, Berlin, 1988.

[18]

W. D. Rong and Y. Ma, ε-Properly efficient solutions of vector optimization problems with set-valued maps, OR Transactions, 4 (2000), 21-32.

[19]

X. M. Yang, D. Li and S. Y. Wang, Near-subconvexlikeness in vector optimization with set-valued functions, J. Optim. Theory Appl., 110 (2001), 413-427. doi: 10.1023/A:1017535631418.

[20]

X. M. Yang, X. Q. Yang and G. Y. Chen, Theorems of the alternative and optimization with set-valued maps, J. Optim. Theory Appl., 107 (2000), 627-640. doi: 10.1023/A:1004613630675.

[21]

K. Q. Zhao and X. M. Yang, E-Benson proper efficiency in vector optimization, Optimization, doi:10.1080/02331934.2013.798321, 2013. doi: 10.1080/02331934.2013.798321.

[22]

K. Q. Zhao, X. M. Yang and J. W. Peng, Weak E-Optimal solution in vector optimization, Taiwan. J. Math., 17 (2013), 1287-1302.

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