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

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

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

Zhiang Zhou ^1, , Xinmin Yang ^2, and Kequan Zhao ^3,

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

Abstract
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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.

Keywords: near $E$-subconvexlikeness, $E$-super efficiency., improvement sets, Set-valued maps.

Mathematics Subject Classification: 90C26, 90C29, 90C3.

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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[3]	Y. H. Cheng and W. T. Fu, Strong efficiency in a locally convex space,, Math. Methods Oper. Res., 50 (1999), 373. 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. 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. 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. Google Scholar
[7]	M. I. Henig, Proper efficiency with respect to cones,, J. Optim. Theory Appl., 36 (1982), 387. 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), 481. Google Scholar
[9]	S. S. Kutateladze, Convex $\epsilon$-programming,, Soviet Math. Dokl., 20 (1979), 391. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. doi: 10.1023/A:1017535631418. Google Scholar
[20]	C. Zălinescu, Convex Analysis in General Vector Spaces,, World Scientific, (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. 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. 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. 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. Google Scholar
[25]	X. Y. Zheng, Proper efficiency in locally convex topological vector spaces,, J. Optim. Theory Appl., 94 (1997), 469. 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. doi: 10.1007/s10957-012-0045-2. Google Scholar

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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. 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. 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. 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. 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. 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. Google Scholar
[7]	M. I. Henig, Proper efficiency with respect to cones,, J. Optim. Theory Appl., 36 (1982), 387. 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), 481. Google Scholar
[9]	S. S. Kutateladze, Convex $\epsilon$-programming,, Soviet Math. Dokl., 20 (1979), 391. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. doi: 10.1023/A:1017535631418. Google Scholar
[20]	C. Zălinescu, Convex Analysis in General Vector Spaces,, World Scientific, (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. 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. 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. 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. Google Scholar
[25]	X. Y. Zheng, Proper efficiency in locally convex topological vector spaces,, J. Optim. Theory Appl., 94 (1997), 469. 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. doi: 10.1007/s10957-012-0045-2. Google Scholar

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