Sensitivity analysis in set-valued optimization under strictly minimal efficiency

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September 2017, 6(3): 427-436. doi: 10.3934/eect.2017022

Sensitivity analysis in set-valued optimization under strictly minimal efficiency

Zhenhua Peng ^1,, , Zhongping Wan ^1, and Weizhi Xiong ^2,

1.	School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China
2.	College of Mathematics and Science, Tongren University, Tongren 554300, China

* Corresponding author: pzhjearya@gmail.com

Received December 2015 Revised May 2017 Published July 2017

Fund Project: The authors are supported by the Natural Science Foundation of China No. 71471140, Natural science program of Guizhou Provincial Department of Education[2015]456 and Collaborative Fund of the Science and Teachnology Department of Guizhou Province[2014]7490

In this paper, the behavior of the perturbation map is analyzed quantitatively by virtue of contingent derivatives and generalized contingent epiderivatives for the set-valued maps under strictly minimal efficiency. The purpose of this paper is to provide some well-known results concerning sensitivity analysis by applying a separation theorem for convex sets. When the results regress to multiobjective optimization, some related conclusions are obtained in a multiobjective programming problem.

Keywords: Sensitivity analysis, perturbation maps, contingent derivatives, generalized contingent epiderivatives, strictly minimal efficiency.

Mathematics Subject Classification: Primary: 90C29, 46G05; Secondary: 49Q12.

Citation: 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

References:

[1]	J. P. Aubin and H. Frankowska, Set-Valued Analysis, Birkhäuser, Boston, 1990.
[2]	J. M. Borwein and D. Zhuang, Super efficiency in vector optimization, Transactions of the American Mathematical Society, 338 (1993), 105-122. doi: 10.1090/S0002-9947-1993-1098432-5.
[3]	G. Y. Chen and J. Jahn, Optimality conditions for set-valued optimization problems, Mathematical Methods of Operations Research, 48 (1998), 187-200. doi: 10.1007/s001860050021.
[4]	Y. H. Cheng and W. T. Fu, Strong efficiency in a locally convex space, Mathematical Methods of Operations Research, 50 (1999), 373-384. doi: 10.1007/s001860050076.
[5]	A. V. Fiacco, Introduction to Sensitivity and Stability Analysis in Nonlinear Programming, Academic Press, New York, 1983.
[6]	W. T. Fu and X. Q. Chen, On approximation families of cones and strictly efficient points, Acta Mathematica Sinica, 40 (1997), 933-938.
[7]	W. T. Fu and Y. H. Cheng, On the strict efficiency in a locally convex space, Journal of Systems Science and Mathematical Sciences, 12 (1999), 40-44.
[8]	X. H. Gong, H. B. Dong and S. Y. Wang, Optimality conditions for proper efficient solutions of vector set-valued optimization, Journal of Mathematical Analysis and Applications, 284 (2003), 332-350. doi: 10.1016/S0022-247X(03)00360-3.
[9]	Y. D. Hu and C. Ling, Connectedness of cone superefficient point sets in locally convex topological vector spaces, Journal of Optimization Theory and Applications, 107 (2000), 433-446. doi: 10.1023/A:1026412918497.
[10]	J. Jahn, Vector Optimization: Theory, Applications, and Extensions, Springer, Berlin Heidelberg, 2004. doi: 10.1007/978-3-540-24828-6.
[11]	S. J. Li, Sensitivity and stability for contingent derivative in multiobjective optimization, Mathematica Applicata, 11 (1998), 49-53.
[12]	Z. F. Li and S. Y. Wang, Connectedness of super efficient sets in vector optimization of set-valued maps, Mathematical Methods of Operations Research, 48 (1998), 207-217. doi: 10.1007/s001860050023.
[13]	R. T. Rockafellar, Lagrange multipliers and subderivatives of optimal value functions in nonlinear programming, Mathematical Programming Study, 17 (1982), 28-66.
[14]	W. D. Rong and Y. N. Wu, $\varepsilon -$weak minimal solution of vector optimization problems with set-valued maps, Journal of Optimization Theory and Applications, 106 (2000), 569-579. doi: 10.1023/A:1004657412928.
[15]	B. H. Sheng and S. Y. Liu, Sensitivity analysis in vector optimization under Benson proper efficiency, Journal of Mathematical Research & Exposition, 22 (2002), 407-412.
[16]	D. S. Shi, Contingent derivative of the perturbation map in multiobjective optimization, Journal of Optimization Theory and Applications, 70 (1991), 385-396. doi: 10.1007/BF00940634.
[17]	D. S. Shi, Sensitivity analysis in convex vector optimization, Journal of Optimization Theory and Applications, 77 (1993), 145-159. doi: 10.1007/BF00940783.
[18]	T. Tanino, Sensitivity analysis in multiobjective optimization, Journal of Optimization Theory and Applications, 56 (1988), 479-499. doi: 10.1007/BF00939554.
[19]	T. Tanino, Stability and sensitivity analysis in convex vector optimization, SIAM Journal on Control and Optimization, 26 (1988), 521-536. doi: 10.1137/0326031.

show all references

References:

[1]	J. P. Aubin and H. Frankowska, Set-Valued Analysis, Birkhäuser, Boston, 1990.
[2]	J. M. Borwein and D. Zhuang, Super efficiency in vector optimization, Transactions of the American Mathematical Society, 338 (1993), 105-122. doi: 10.1090/S0002-9947-1993-1098432-5.
[3]	G. Y. Chen and J. Jahn, Optimality conditions for set-valued optimization problems, Mathematical Methods of Operations Research, 48 (1998), 187-200. doi: 10.1007/s001860050021.
[4]	Y. H. Cheng and W. T. Fu, Strong efficiency in a locally convex space, Mathematical Methods of Operations Research, 50 (1999), 373-384. doi: 10.1007/s001860050076.
[5]	A. V. Fiacco, Introduction to Sensitivity and Stability Analysis in Nonlinear Programming, Academic Press, New York, 1983.
[6]	W. T. Fu and X. Q. Chen, On approximation families of cones and strictly efficient points, Acta Mathematica Sinica, 40 (1997), 933-938.
[7]	W. T. Fu and Y. H. Cheng, On the strict efficiency in a locally convex space, Journal of Systems Science and Mathematical Sciences, 12 (1999), 40-44.
[8]	X. H. Gong, H. B. Dong and S. Y. Wang, Optimality conditions for proper efficient solutions of vector set-valued optimization, Journal of Mathematical Analysis and Applications, 284 (2003), 332-350. doi: 10.1016/S0022-247X(03)00360-3.
[9]	Y. D. Hu and C. Ling, Connectedness of cone superefficient point sets in locally convex topological vector spaces, Journal of Optimization Theory and Applications, 107 (2000), 433-446. doi: 10.1023/A:1026412918497.
[10]	J. Jahn, Vector Optimization: Theory, Applications, and Extensions, Springer, Berlin Heidelberg, 2004. doi: 10.1007/978-3-540-24828-6.
[11]	S. J. Li, Sensitivity and stability for contingent derivative in multiobjective optimization, Mathematica Applicata, 11 (1998), 49-53.
[12]	Z. F. Li and S. Y. Wang, Connectedness of super efficient sets in vector optimization of set-valued maps, Mathematical Methods of Operations Research, 48 (1998), 207-217. doi: 10.1007/s001860050023.
[13]	R. T. Rockafellar, Lagrange multipliers and subderivatives of optimal value functions in nonlinear programming, Mathematical Programming Study, 17 (1982), 28-66.
[14]	W. D. Rong and Y. N. Wu, $\varepsilon -$weak minimal solution of vector optimization problems with set-valued maps, Journal of Optimization Theory and Applications, 106 (2000), 569-579. doi: 10.1023/A:1004657412928.
[15]	B. H. Sheng and S. Y. Liu, Sensitivity analysis in vector optimization under Benson proper efficiency, Journal of Mathematical Research & Exposition, 22 (2002), 407-412.
[16]	D. S. Shi, Contingent derivative of the perturbation map in multiobjective optimization, Journal of Optimization Theory and Applications, 70 (1991), 385-396. doi: 10.1007/BF00940634.
[17]	D. S. Shi, Sensitivity analysis in convex vector optimization, Journal of Optimization Theory and Applications, 77 (1993), 145-159. doi: 10.1007/BF00940783.
[18]	T. Tanino, Sensitivity analysis in multiobjective optimization, Journal of Optimization Theory and Applications, 56 (1988), 479-499. doi: 10.1007/BF00939554.
[19]	T. Tanino, Stability and sensitivity analysis in convex vector optimization, SIAM Journal on Control and Optimization, 26 (1988), 521-536. doi: 10.1137/0326031.

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