# American Institute of Mathematical Sciences

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.

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##### 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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