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:
[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.

[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.

[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.

[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.

[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.

[6]

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

[7]

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

[8]

H. Kuhn and A. Tucker, Nonlinear programming,, in Proceedings of the 2nd Berkeley Symposium on Mathematical Statistics and Probability, (1950), 481.

[9]

S. S. Kutateladze, Convex $\epsilon$-programming,, Soviet Math. Dokl., 20 (1979), 391.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[20]

C. Zălinescu, Convex Analysis in General Vector Spaces,, World Scientific, (2002). doi: 10.1142/9789812777096.

[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.

[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.

[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.

[24]

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

[25]

X. Y. Zheng, Proper efficiency in locally convex topological vector spaces,, J. Optim. Theory Appl., 94 (1997), 469. doi: 10.1023/A:1022648115446.

[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.

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. doi: 10.1016/0022-247X(79)90226-9.

[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.

[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.

[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.

[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.

[6]

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

[7]

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

[8]

H. Kuhn and A. Tucker, Nonlinear programming,, in Proceedings of the 2nd Berkeley Symposium on Mathematical Statistics and Probability, (1950), 481.

[9]

S. S. Kutateladze, Convex $\epsilon$-programming,, Soviet Math. Dokl., 20 (1979), 391.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[20]

C. Zălinescu, Convex Analysis in General Vector Spaces,, World Scientific, (2002). doi: 10.1142/9789812777096.

[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.

[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.

[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.

[24]

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

[25]

X. Y. Zheng, Proper efficiency in locally convex topological vector spaces,, J. Optim. Theory Appl., 94 (1997), 469. doi: 10.1023/A:1022648115446.

[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.

[1]

Guolin Yu. Global proper efficiency and vector optimization with cone-arcwise connected set-valued maps. Numerical Algebra, Control & Optimization, 2016, 6 (1) : 35-44. doi: 10.3934/naco.2016.6.35

[2]

Dante Carrasco-Olivera, Roger Metzger Alvan, Carlos Arnoldo Morales Rojas. Topological entropy for set-valued maps. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3461-3474. doi: 10.3934/dcdsb.2015.20.3461

[3]

Zhenhua Peng, Zhongping Wan, Weizhi Xiong. Sensitivity analysis in set-valued optimization under strictly minimal efficiency. Evolution Equations & Control Theory, 2017, 6 (3) : 427-436. doi: 10.3934/eect.2017022

[4]

Yihong Xu, Zhenhua Peng. Higher-order sensitivity analysis in set-valued optimization under Henig efficiency. Journal of Industrial & Management Optimization, 2017, 13 (1) : 313-327. doi: 10.3934/jimo.2016019

[5]

Ying Gao, Xinmin Yang, Jin Yang, Hong Yan. Scalarizations and Lagrange multipliers for approximate solutions in the vector optimization problems with set-valued maps. Journal of Industrial & Management Optimization, 2015, 11 (2) : 673-683. doi: 10.3934/jimo.2015.11.673

[6]

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

[7]

Roger Metzger, Carlos Arnoldo Morales Rojas, Phillipe Thieullen. Topological stability in set-valued dynamics. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1965-1975. doi: 10.3934/dcdsb.2017115

[8]

Geng-Hua Li, Sheng-Jie Li. Unified optimality conditions for set-valued optimizations. Journal of Industrial & Management Optimization, 2019, 15 (3) : 1101-1116. doi: 10.3934/jimo.2018087

[9]

Tao Chen, Yunping Jiang, Gaofei Zhang. No invariant line fields on escaping sets of the family $\lambda e^{iz}+\gamma e^{-iz}$. Discrete & Continuous Dynamical Systems - A, 2013, 33 (5) : 1883-1890. doi: 10.3934/dcds.2013.33.1883

[10]

Yu Zhang, Tao Chen. Minimax problems for set-valued mappings with set optimization. Numerical Algebra, Control & Optimization, 2014, 4 (4) : 327-340. doi: 10.3934/naco.2014.4.327

[11]

Caili Sang, Zhen Chen. $ E $-eigenvalue localization sets for tensors. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-19. doi: 10.3934/jimo.2019042

[12]

Qingbang Zhang, Caozong Cheng, Xuanxuan Li. Generalized minimax theorems for two set-valued mappings. Journal of Industrial & Management Optimization, 2013, 9 (1) : 1-12. doi: 10.3934/jimo.2013.9.1

[13]

Sina Greenwood, Rolf Suabedissen. 2-manifolds and inverse limits of set-valued functions on intervals. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5693-5706. doi: 10.3934/dcds.2017246

[14]

Mariusz Michta. Stochastic inclusions with non-continuous set-valued operators. Conference Publications, 2009, 2009 (Special) : 548-557. doi: 10.3934/proc.2009.2009.548

[15]

Guolin Yu. Topological properties of Henig globally efficient solutions of set-valued problems. Numerical Algebra, Control & Optimization, 2014, 4 (4) : 309-316. doi: 10.3934/naco.2014.4.309

[16]

Zengjing Chen, Yuting Lan, Gaofeng Zong. Strong law of large numbers for upper set-valued and fuzzy-set valued probability. Mathematical Control & Related Fields, 2015, 5 (3) : 435-452. doi: 10.3934/mcrf.2015.5.435

[17]

C. R. Chen, S. J. Li. Semicontinuity of the solution set map to a set-valued weak vector variational inequality. Journal of Industrial & Management Optimization, 2007, 3 (3) : 519-528. doi: 10.3934/jimo.2007.3.519

[18]

Jiawei Chen, Zhongping Wan, Liuyang Yuan. Existence of solutions and $\alpha$-well-posedness for a system of constrained set-valued variational inequalities. Numerical Algebra, Control & Optimization, 2013, 3 (3) : 567-581. doi: 10.3934/naco.2013.3.567

[19]

Benjamin Seibold, Morris R. Flynn, Aslan R. Kasimov, Rodolfo R. Rosales. Constructing set-valued fundamental diagrams from Jamiton solutions in second order traffic models. Networks & Heterogeneous Media, 2013, 8 (3) : 745-772. doi: 10.3934/nhm.2013.8.745

[20]

Shay Kels, Nira Dyn. Bernstein-type approximation of set-valued functions in the symmetric difference metric. Discrete & Continuous Dynamical Systems - A, 2014, 34 (3) : 1041-1060. doi: 10.3934/dcds.2014.34.1041

2017 Impact Factor: 0.994

Metrics

  • PDF downloads (13)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]