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February  2017, 13(1): 313-327. doi: 10.3934/jimo.2016019

Higher-order sensitivity analysis in set-valued optimization under Henig efficiency

Department of Mathematics, Nanchang University, Nanchang, 330031, China

Yihong Xu, Professor, major field of interest is in the area of set-valued optimization. E-mail: xuyihong@ncu.edu.cn

Zhenhua Peng, E-mail: pzhjearya@gmail.com

Received  April 2015 Revised  December 2015 Published  March 2016

Fund Project: the National Natural Science Foundation of China Grant 11461044, the Natural Science Foundation of Jiangxi Province (20151BAB201027) and the Science and Technology Foundation of the Education Department of Jiangxi Province(GJJ12010).

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.

Citation: Yihong Xu, Zhenhua Peng. Higher-order sensitivity analysis in set-valued optimization under Henig efficiency. Journal of Industrial & Management Optimization, 2017, 13 (1) : 313-327. doi: 10.3934/jimo.2016019
References:
[1] J.P. Aubin and H. Frankowska, Set-valued Analysis, Birkhäuser, Boston, 1990.   Google Scholar
[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.   Google Scholar

[3]

J.M. Borwein and D. Zhuang, Super efficiency in vector optimization, Transactions of the American Mathematical Society, 338 (1993), 105-122.   Google Scholar

[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.   Google Scholar

[5] A.V. Fiacco, Introduction to Sensitivity and Stability Analysis in Nonlinear Programming, Academic Press. New York, 1983.   Google Scholar
[6]

M.I. Henig, Proper efficiency with respect to cones, Journal of Optimization Theory and Applications, 36 (1982), 387-407.   Google Scholar

[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.   Google Scholar

[8] J. Jahn, Vector Optimization: Theory, Springer Berlin Heidel-berg, 2004.   Google Scholar
[9]

S.J. Li, Sensitivity and stability for contingent derivative in multiobjective optimization, Mathematica Applicata, 11 (1998), 49-53.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[12]

S. J. LiX. 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.   Google Scholar

[13]

X. B. LiX. 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.   Google Scholar

[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.   Google Scholar

[15]

R. T. Rockafellar, Lagrange multipliers and subderivatives of optimal value functions in non-linear programming, Mathematical Programming Study, 17 (1982), 28-66.   Google Scholar

[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.   Google Scholar

[17]

D. S. Shi, Contingent derivative of the perturbation in multiobjective optimization, Journal of Optimization Theory and Applications, 70 (1991), 351-362.   Google Scholar

[18]

D. S. Shi, Sensitivity analysis in convex vector optimization, Journal of Optimization Theory and Applications, 77 (1993), 145-159.   Google Scholar

[19]

T. Tanino, Sensitivity analysis in multiobjective optimization, Journal of Optimization Theory and Applications, 56 (1988), 479-499.   Google Scholar

[20]

T. Tanino, Stability and sensitivity analysis in convex vector optimization, SIAM Journal on Control and Optimization, 26 (1988), 521-536.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[23]

X. Y. Zheng, Proper efficiency in locally convex topological vector spaces, Journal of Opti-mization Theory and Applications, 94 (1997), 469-486.   Google Scholar

show all references

References:
[1] J.P. Aubin and H. Frankowska, Set-valued Analysis, Birkhäuser, Boston, 1990.   Google Scholar
[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.   Google Scholar

[3]

J.M. Borwein and D. Zhuang, Super efficiency in vector optimization, Transactions of the American Mathematical Society, 338 (1993), 105-122.   Google Scholar

[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.   Google Scholar

[5] A.V. Fiacco, Introduction to Sensitivity and Stability Analysis in Nonlinear Programming, Academic Press. New York, 1983.   Google Scholar
[6]

M.I. Henig, Proper efficiency with respect to cones, Journal of Optimization Theory and Applications, 36 (1982), 387-407.   Google Scholar

[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.   Google Scholar

[8] J. Jahn, Vector Optimization: Theory, Springer Berlin Heidel-berg, 2004.   Google Scholar
[9]

S.J. Li, Sensitivity and stability for contingent derivative in multiobjective optimization, Mathematica Applicata, 11 (1998), 49-53.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[12]

S. J. LiX. 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.   Google Scholar

[13]

X. B. LiX. 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.   Google Scholar

[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.   Google Scholar

[15]

R. T. Rockafellar, Lagrange multipliers and subderivatives of optimal value functions in non-linear programming, Mathematical Programming Study, 17 (1982), 28-66.   Google Scholar

[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.   Google Scholar

[17]

D. S. Shi, Contingent derivative of the perturbation in multiobjective optimization, Journal of Optimization Theory and Applications, 70 (1991), 351-362.   Google Scholar

[18]

D. S. Shi, Sensitivity analysis in convex vector optimization, Journal of Optimization Theory and Applications, 77 (1993), 145-159.   Google Scholar

[19]

T. Tanino, Sensitivity analysis in multiobjective optimization, Journal of Optimization Theory and Applications, 56 (1988), 479-499.   Google Scholar

[20]

T. Tanino, Stability and sensitivity analysis in convex vector optimization, SIAM Journal on Control and Optimization, 26 (1988), 521-536.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[23]

X. Y. Zheng, Proper efficiency in locally convex topological vector spaces, Journal of Opti-mization Theory and Applications, 94 (1997), 469-486.   Google Scholar

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