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Sensitivity analysis in set-valued optimization under strictly minimal efficiency
1. | School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China |
2. | College of Mathematics and Science, Tongren University, Tongren 554300, China |
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.
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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