American Institute of Mathematical Sciences

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

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

Kequan Zhao 1, and  Xinmin Yang 1,

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.
Keywords: duality., Vector optimization, saddle point criteria, scalarization, $E$-Benson proper efficiency, Lagrange multipliers, nearly $E$-Subconvexlike-ness.
Mathematics Subject Classification: Primary: 49N15; Secondary: 90C26, 90C29, 90C4.
Citation: Kequan Zhao, Xinmin Yang. Characterizations of the $E$-Benson proper efficiency in vector optimization problems. Numerical Algebra, Control & 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.  Google Scholar

[2]

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

[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.  Google Scholar

[4]

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

[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.  Google Scholar

[6]

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

[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.  Google Scholar

[8]

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

[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.  Google Scholar

[10]

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

[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. Google Scholar

[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.  Google Scholar

[13]

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

[14]

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

[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.  Google Scholar

[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.  Google Scholar

[17]

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

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

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.  Google Scholar

[2]

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

[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.  Google Scholar

[4]

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

[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.  Google Scholar

[6]

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

[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.  Google Scholar

[8]

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

[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.  Google Scholar

[10]

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

[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. Google Scholar

[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.  Google Scholar

[13]

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

[14]

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

[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.  Google Scholar

[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.  Google Scholar

[17]

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

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

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