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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 and 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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