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Higher-order sensitivity analysis in set-valued optimization under Henig efficiency
Department of Mathematics, Nanchang University, Nanchang, 330031, China |
The behavior of the perturbation map is analyzed quantitatively by using the concept of higher-order contingent derivative for the set-valued maps under Henig efficiency. By using the higher-order contingent derivatives and applying a separation theorem for convex sets, some results concerning higher-order sensitivity analysis are established.
References:
| [1] |
J.P. Aubin and H. Frankowska, Set-valued Analysis, Birkhäuser, Boston, 1990.
|
| [2] |
H.P. Benson,
An improved denition of proper efficiency for vector maximization with respect to cones, Journal of Mathematical Analysis and Applications, 71 (1979), 232-241.
|
| [3] |
J.M. Borwein and D. Zhuang,
Super efficiency in vector optimization, Transactions of the American Mathematical Society, 338 (1993), 105-122.
|
| [4] |
H.Y. Deng and W. Wei,
Existence and stability analysis for nonlinear optimal control problems with 1-mean equicontinuous controls, Journal of Industrial and Management Optimization, 11 (2015), 1409-1422.
|
| [5] |
A.V. Fiacco, Introduction to Sensitivity and Stability Analysis in Nonlinear Programming, Academic Press. New York, 1983.
|
| [6] |
M.I. Henig,
Proper efficiency with respect to cones, Journal of Optimization Theory and Applications, 36 (1982), 387-407.
|
| [7] |
Y.D. Hu and C. Ling,
Connectedness of cone superefficient point sets in locally convex topo-logical vector spaces, Journal of Optimization Theory and Applications, 107 (2000), 433-446.
|
| [8] |
J. Jahn, Vector Optimization: Theory, Springer Berlin Heidel-berg, 2004.
|
| [9] |
S.J. Li,
Sensitivity and stability for contingent derivative in multiobjective optimization, Mathematica Applicata, 11 (1998), 49-53.
|
| [10] |
S. J. Li and C. R. Chen,
Higher order optimality conditions for henig efficient solutions in set-valued optimization, Journal of Mathematical Analysis and Applications, 323 (2006), 1184-1200.
|
| [11] |
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.
|
| [12] |
S. J. Li, X. K. Sun and J. Zhai,
Second-order contingent derivatives of set-valued mappings with application to set-valued optimization, Applied Mathematics and Computation, 218 (2012), 6874-6886.
|
| [13] |
X. B. Li, X. J. Long and Z. Lin,
Stability of solution mapping for parametric symmetric vector equilibrium problems, Journal of Industrial and Management Optimization, 11 (2015), 661-671.
|
| [14] |
Q. S. Qiu and X. M. Yang,
Connectedness of Henig weakly efficient solution set for set-valued optimization problems, Journal of Optimization Theory and Applications, 152 (2012), 439-449.
|
| [15] |
R. T. Rockafellar,
Lagrange multipliers and subderivatives of optimal value functions in non-linear programming, Mathematical Programming Study, 17 (1982), 28-66.
|
| [16] |
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.
|
| [17] |
D. S. Shi,
Contingent derivative of the perturbation in multiobjective optimization, Journal of Optimization Theory and Applications, 70 (1991), 351-362.
|
| [18] |
D. S. Shi,
Sensitivity analysis in convex vector optimization, Journal of Optimization Theory and Applications, 77 (1993), 145-159.
|
| [19] |
T. Tanino,
Sensitivity analysis in multiobjective optimization, Journal of Optimization Theory and Applications, 56 (1988), 479-499.
|
| [20] |
T. Tanino,
Stability and sensitivity analysis in convex vector optimization, SIAM Journal on Control and Optimization, 26 (1988), 521-536.
|
| [21] |
Q. L. Wang and S. J. Li,
Generalized higher-order optimality conditions for set-valued opti-mization under Henig efficiency, Numerical Functional Analysis and Optimization, 30 (2009), 849-869.
|
| [22] |
D. E. Ward,
A chain rule for first and second order epiderivatives and hypoderivatives, Journal of Mathematical Analysis and Applications, 348 (2008), 324-336.
|
| [23] |
X. Y. Zheng,
Proper efficiency in locally convex topological vector spaces, Journal of Opti-mization Theory and Applications, 94 (1997), 469-486.
|
show all references
References:
| [1] |
J.P. Aubin and H. Frankowska, Set-valued Analysis, Birkhäuser, Boston, 1990.
|
| [2] |
H.P. Benson,
An improved denition of proper efficiency for vector maximization with respect to cones, Journal of Mathematical Analysis and Applications, 71 (1979), 232-241.
|
| [3] |
J.M. Borwein and D. Zhuang,
Super efficiency in vector optimization, Transactions of the American Mathematical Society, 338 (1993), 105-122.
|
| [4] |
H.Y. Deng and W. Wei,
Existence and stability analysis for nonlinear optimal control problems with 1-mean equicontinuous controls, Journal of Industrial and Management Optimization, 11 (2015), 1409-1422.
|
| [5] |
A.V. Fiacco, Introduction to Sensitivity and Stability Analysis in Nonlinear Programming, Academic Press. New York, 1983.
|
| [6] |
M.I. Henig,
Proper efficiency with respect to cones, Journal of Optimization Theory and Applications, 36 (1982), 387-407.
|
| [7] |
Y.D. Hu and C. Ling,
Connectedness of cone superefficient point sets in locally convex topo-logical vector spaces, Journal of Optimization Theory and Applications, 107 (2000), 433-446.
|
| [8] |
J. Jahn, Vector Optimization: Theory, Springer Berlin Heidel-berg, 2004.
|
| [9] |
S.J. Li,
Sensitivity and stability for contingent derivative in multiobjective optimization, Mathematica Applicata, 11 (1998), 49-53.
|
| [10] |
S. J. Li and C. R. Chen,
Higher order optimality conditions for henig efficient solutions in set-valued optimization, Journal of Mathematical Analysis and Applications, 323 (2006), 1184-1200.
|
| [11] |
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.
|
| [12] |
S. J. Li, X. K. Sun and J. Zhai,
Second-order contingent derivatives of set-valued mappings with application to set-valued optimization, Applied Mathematics and Computation, 218 (2012), 6874-6886.
|
| [13] |
X. B. Li, X. J. Long and Z. Lin,
Stability of solution mapping for parametric symmetric vector equilibrium problems, Journal of Industrial and Management Optimization, 11 (2015), 661-671.
|
| [14] |
Q. S. Qiu and X. M. Yang,
Connectedness of Henig weakly efficient solution set for set-valued optimization problems, Journal of Optimization Theory and Applications, 152 (2012), 439-449.
|
| [15] |
R. T. Rockafellar,
Lagrange multipliers and subderivatives of optimal value functions in non-linear programming, Mathematical Programming Study, 17 (1982), 28-66.
|
| [16] |
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.
|
| [17] |
D. S. Shi,
Contingent derivative of the perturbation in multiobjective optimization, Journal of Optimization Theory and Applications, 70 (1991), 351-362.
|
| [18] |
D. S. Shi,
Sensitivity analysis in convex vector optimization, Journal of Optimization Theory and Applications, 77 (1993), 145-159.
|
| [19] |
T. Tanino,
Sensitivity analysis in multiobjective optimization, Journal of Optimization Theory and Applications, 56 (1988), 479-499.
|
| [20] |
T. Tanino,
Stability and sensitivity analysis in convex vector optimization, SIAM Journal on Control and Optimization, 26 (1988), 521-536.
|
| [21] |
Q. L. Wang and S. J. Li,
Generalized higher-order optimality conditions for set-valued opti-mization under Henig efficiency, Numerical Functional Analysis and Optimization, 30 (2009), 849-869.
|
| [22] |
D. E. Ward,
A chain rule for first and second order epiderivatives and hypoderivatives, Journal of Mathematical Analysis and Applications, 348 (2008), 324-336.
|
| [23] |
X. Y. Zheng,
Proper efficiency in locally convex topological vector spaces, Journal of Opti-mization Theory and Applications, 94 (1997), 469-486.
|
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