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Higherorder sensitivity analysis in setvalued 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 higherorder contingent derivative for the setvalued maps under Henig efficiency. By using the higherorder contingent derivatives and applying a separation theorem for convex sets, some results concerning higherorder sensitivity analysis are established.
References:
[1]  J.P. Aubin and H. Frankowska, Setvalued 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), 232241. 
[3] 
J.M. Borwein and D. Zhuang, Super efficiency in vector optimization, Transactions of the American Mathematical Society, 338 (1993), 105122. 
[4] 
H.Y. Deng and W. Wei, Existence and stability analysis for nonlinear optimal control problems with 1mean equicontinuous controls, Journal of Industrial and Management Optimization, 11 (2015), 14091422. 
[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), 387407. 
[7] 
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), 433446. 
[8]  J. Jahn, Vector Optimization: Theory, Springer Berlin Heidelberg, 2004. 
[9] 
S.J. Li, Sensitivity and stability for contingent derivative in multiobjective optimization, Mathematica Applicata, 11 (1998), 4953. 
[10] 
S. J. Li and C. R. Chen, Higher order optimality conditions for henig efficient solutions in setvalued optimization, Journal of Mathematical Analysis and Applications, 323 (2006), 11841200. 
[11] 
Z. F. Li and S. Y. Wang, Connectedness of super efficient sets in vector optimization of setvalued maps, Mathematical Methods of Operations Research, 48 (1998), 207217. 
[12] 
S. J. Li, X. K. Sun and J. Zhai, Secondorder contingent derivatives of setvalued mappings with application to setvalued optimization, Applied Mathematics and Computation, 218 (2012), 68746886. 
[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), 661671. 
[14] 
Q. S. Qiu and X. M. Yang, Connectedness of Henig weakly efficient solution set for setvalued optimization problems, Journal of Optimization Theory and Applications, 152 (2012), 439449. 
[15] 
R. T. Rockafellar, Lagrange multipliers and subderivatives of optimal value functions in nonlinear programming, Mathematical Programming Study, 17 (1982), 2866. 
[16] 
B. H. Sheng and S. Y. Liu, Sensitivity analysis in vector optimization under Benson proper efficiency, Journal of Mathematical Research & Exposition, 22 (2002), 407412. 
[17] 
D. S. Shi, Contingent derivative of the perturbation in multiobjective optimization, Journal of Optimization Theory and Applications, 70 (1991), 351362. 
[18] 
D. S. Shi, Sensitivity analysis in convex vector optimization, Journal of Optimization Theory and Applications, 77 (1993), 145159. 
[19] 
T. Tanino, Sensitivity analysis in multiobjective optimization, Journal of Optimization Theory and Applications, 56 (1988), 479499. 
[20] 
T. Tanino, Stability and sensitivity analysis in convex vector optimization, SIAM Journal on Control and Optimization, 26 (1988), 521536. 
[21] 
Q. L. Wang and S. J. Li, Generalized higherorder optimality conditions for setvalued optimization under Henig efficiency, Numerical Functional Analysis and Optimization, 30 (2009), 849869. 
[22] 
D. E. Ward, A chain rule for first and second order epiderivatives and hypoderivatives, Journal of Mathematical Analysis and Applications, 348 (2008), 324336. 
[23] 
X. Y. Zheng, Proper efficiency in locally convex topological vector spaces, Journal of Optimization Theory and Applications, 94 (1997), 469486. 
show all references
References:
[1]  J.P. Aubin and H. Frankowska, Setvalued 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), 232241. 
[3] 
J.M. Borwein and D. Zhuang, Super efficiency in vector optimization, Transactions of the American Mathematical Society, 338 (1993), 105122. 
[4] 
H.Y. Deng and W. Wei, Existence and stability analysis for nonlinear optimal control problems with 1mean equicontinuous controls, Journal of Industrial and Management Optimization, 11 (2015), 14091422. 
[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), 387407. 
[7] 
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), 433446. 
[8]  J. Jahn, Vector Optimization: Theory, Springer Berlin Heidelberg, 2004. 
[9] 
S.J. Li, Sensitivity and stability for contingent derivative in multiobjective optimization, Mathematica Applicata, 11 (1998), 4953. 
[10] 
S. J. Li and C. R. Chen, Higher order optimality conditions for henig efficient solutions in setvalued optimization, Journal of Mathematical Analysis and Applications, 323 (2006), 11841200. 
[11] 
Z. F. Li and S. Y. Wang, Connectedness of super efficient sets in vector optimization of setvalued maps, Mathematical Methods of Operations Research, 48 (1998), 207217. 
[12] 
S. J. Li, X. K. Sun and J. Zhai, Secondorder contingent derivatives of setvalued mappings with application to setvalued optimization, Applied Mathematics and Computation, 218 (2012), 68746886. 
[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), 661671. 
[14] 
Q. S. Qiu and X. M. Yang, Connectedness of Henig weakly efficient solution set for setvalued optimization problems, Journal of Optimization Theory and Applications, 152 (2012), 439449. 
[15] 
R. T. Rockafellar, Lagrange multipliers and subderivatives of optimal value functions in nonlinear programming, Mathematical Programming Study, 17 (1982), 2866. 
[16] 
B. H. Sheng and S. Y. Liu, Sensitivity analysis in vector optimization under Benson proper efficiency, Journal of Mathematical Research & Exposition, 22 (2002), 407412. 
[17] 
D. S. Shi, Contingent derivative of the perturbation in multiobjective optimization, Journal of Optimization Theory and Applications, 70 (1991), 351362. 
[18] 
D. S. Shi, Sensitivity analysis in convex vector optimization, Journal of Optimization Theory and Applications, 77 (1993), 145159. 
[19] 
T. Tanino, Sensitivity analysis in multiobjective optimization, Journal of Optimization Theory and Applications, 56 (1988), 479499. 
[20] 
T. Tanino, Stability and sensitivity analysis in convex vector optimization, SIAM Journal on Control and Optimization, 26 (1988), 521536. 
[21] 
Q. L. Wang and S. J. Li, Generalized higherorder optimality conditions for setvalued optimization under Henig efficiency, Numerical Functional Analysis and Optimization, 30 (2009), 849869. 
[22] 
D. E. Ward, A chain rule for first and second order epiderivatives and hypoderivatives, Journal of Mathematical Analysis and Applications, 348 (2008), 324336. 
[23] 
X. Y. Zheng, Proper efficiency in locally convex topological vector spaces, Journal of Optimization Theory and Applications, 94 (1997), 469486. 
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