# American Institute of Mathematical Sciences

July  2016, 12(3): 1031-1039. doi: 10.3934/jimo.2016.12.1031

## $E$-super efficiency of set-valued optimization problems involving improvement sets

 1 College of Mathematics and Statistics, Chongqing University of Technology, Chongqing 400054, China 2 Department of Mathematics, Chongqing Normal University, Chongqing 400047 3 College of Mathematics Science, Chongqing normal University, Chongqing 400047, China

Received  November 2014 Revised  May 2015 Published  September 2015

In this paper, $E$-super efficiency of set-valued optimization problems is investigated. Firstly, based on the improvement set, a new notion of $E$-super efficient point is introduced in real locally convex spaces. Secondly, under the assumption of near $E$-subconvexlikeness of set-valued maps, scalarization theorems of set-valued optimization problems are established in the sense of $E$-super efficiency. Finally, Lagrange multiplier theorems of set-valued optimization problems are obtained in the sense of $E$-super efficiency.
Citation: Zhiang Zhou, Xinmin Yang, Kequan Zhao. $E$-super efficiency of set-valued optimization problems involving improvement sets. Journal of Industrial & Management Optimization, 2016, 12 (3) : 1031-1039. doi: 10.3934/jimo.2016.12.1031
##### References:
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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.  Google Scholar [2] J. M. Borwein and D. M. Zhuang, Super efficiency in vector optimization, Trans. Am. Math. Soc., 338 (1993), 105-122. doi: 10.1090/S0002-9947-1993-1098432-5.  Google Scholar [3] Y. H. Cheng and W. T. Fu, Strong efficiency in a locally convex space, Math. Methods Oper. Res., 50 (1999), 373-384. doi: 10.1007/s001860050076.  Google Scholar [4] 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 [5] A. M. Geoffrion, Proper efficiency and the theory of vector maximization, J. Math. Anal. Appl., 22 (1968), 618-630. doi: 10.1016/0022-247X(68)90201-1.  Google Scholar [6] C. Gutiérrez, B. Jiménez and V. Novo, Improvement sets and vector optimization, Eur. J. Oper. Res., 223 (2012), 304-311. Google Scholar [7] M. I. Henig, Proper efficiency with respect to cones, J. Optim. Theory Appl., 36 (1982), 387-407. doi: 10.1007/BF00934353.  Google Scholar [8] H. Kuhn and A. Tucker, Nonlinear programming, in Proceedings of the 2nd Berkeley Symposium on Mathematical Statistics and Probability, 1950, University of California Press, Berkeley, 1951, 481-492.  Google Scholar [9] S. S. Kutateladze, Convex $\epsilon$-programming, Soviet Math. Dokl., 20 (1979), 391-393. Google Scholar [10] T. Y. Li, Y. H. Xu and C. X. Zhu, $\varepsilon$-Strictly efficient solutions of vector optimization problems with set-valued maps, Asia. Pacific. J. Oper. Res., 24 (2007), 841-854. doi: 10.1142/S0217595907001577.  Google Scholar [11] Z. M. 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 [12] A. Mehra, Super efficiency in vector optimization with nearly convexlike set-valued maps, J. Math. Anal. Appl., 276 (2002), 815-832. doi: 10.1016/S0022-247X(02)00452-3.  Google Scholar [13] Q. S. Qiu and W. T. Fu, The connectedness of the super efficient solution sets of the optimization problem for a set-valued mapping, J. Sys. Sci. & Math. Scis., 22 (2002), 107-114.  Google Scholar [14] W. D. Rong and Y. N. Wu, Characterizations of super efficiency in cone-convexlike vector optimization with set-valued maps, Math. Methods Oper. Res., 48 (1998), 247-258. doi: 10.1007/s001860050026.  Google Scholar [15] W. D. Rong and Y. N. Wu, $\epsilon$-Weak minimal solutions of vector optimization problems with set-valued maps, J. Optim. Theory Appl., 106 (2000), 569-579. doi: 10.1023/A:1004657412928.  Google Scholar [16] L. A. Tuan, $\varepsilon$-Optimality conditions for vector optimization problems with set-valued maps, Numer. Func. Anal. Optim., 31 (2010), 78-95. doi: 10.1080/01630560903499845.  Google Scholar [17] L. Y. Xia and J. H. Qiu, Superefficiency in vector optimization with nearly subconvexlike set-valued maps, J. Optim. Theory Appl., 136 (2008), 125-137. doi: 10.1007/s10957-007-9291-0.  Google Scholar [18] Y. H. Xu and S. Y. Liu, Super efficiency in the nearly cone-subconvexlike vector optimization with set-valued functions, Acta. Math. Sci. B, 25 (2005), 152-160.  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] C. Zălinescu, Convex Analysis in General Vector Spaces, World Scientific, New York, 2002. doi: 10.1142/9789812777096.  Google Scholar [21] K. Q. Zhao and X. M. Yang, $E$-proper saddle points and $E$-proper duality in vector optimization with set-valued maps, Taiwan. J. Math., 18 (2014), 483-495. doi: 10.11650/tjm.18.2014.3473.  Google Scholar [22] K. Q. Zhao and X. M. Yang, $E$-Benson proper efficiency in vector optimization, Optimization, 64 (2015), 739-752. doi: 10.1080/02331934.2013.798321.  Google Scholar [23] K. Q. Zhao and X. M. Yang, Characterizations of the $E$-Benson proper efficiency in vector optimization problems, Numer. Algebr. Control. Optim., 3 (2013), 643-653. doi: 10.3934/naco.2013.3.643.  Google Scholar [24] 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 [25] X. Y. Zheng, Proper efficiency in locally convex topological vector spaces, J. Optim. Theory Appl., 94 (1997), 469-486. doi: 10.1023/A:1022648115446.  Google Scholar [26] Z. A. Zhou and J. W. Peng, Scalarization of set-valued optimization problems with generalization cone subconvexlikeness in real ordered linear spaces, J. Optim. Theory Appl., 154 (2012), 830-841. doi: 10.1007/s10957-012-0045-2.  Google Scholar
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