American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Long-time dynamics for a class of extensible beams with nonlocal nonlinear damping*
  • EECT Home
  • This Issue
  • Next Article
    $\mathbb{L}^p-$solutions of the stochastic Navier-Stokes equations subject to Lévy noise with $\mathbb{L}^m(\mathbb{R}^m)$ initial data
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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

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

[6]

W. T. Fu and X. Q. Chen, On approximation families of cones and strictly efficient points, Acta Mathematica Sinica, 40 (1997), 933-938.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

J. Jahn, Vector Optimization: Theory, Applications, and Extensions, Springer, Berlin Heidelberg, 2004. doi: 10.1007/978-3-540-24828-6.  Google Scholar

[11]

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

[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.  Google Scholar

[13]

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

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[17]

D. S. Shi, Sensitivity analysis in convex vector optimization, Journal of Optimization Theory and Applications, 77 (1993), 145-159.  doi: 10.1007/BF00940783.  Google Scholar

[18]

T. Tanino, Sensitivity analysis in multiobjective optimization, Journal of Optimization Theory and Applications, 56 (1988), 479-499.  doi: 10.1007/BF00939554.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

J. P. Aubin and H. Frankowska, Set-Valued Analysis, Birkhäuser, Boston, 1990.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

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

[6]

W. T. Fu and X. Q. Chen, On approximation families of cones and strictly efficient points, Acta Mathematica Sinica, 40 (1997), 933-938.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

J. Jahn, Vector Optimization: Theory, Applications, and Extensions, Springer, Berlin Heidelberg, 2004. doi: 10.1007/978-3-540-24828-6.  Google Scholar

[11]

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

[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.  Google Scholar

[13]

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

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[17]

D. S. Shi, Sensitivity analysis in convex vector optimization, Journal of Optimization Theory and Applications, 77 (1993), 145-159.  doi: 10.1007/BF00940783.  Google Scholar

[18]

T. Tanino, Sensitivity analysis in multiobjective optimization, Journal of Optimization Theory and Applications, 56 (1988), 479-499.  doi: 10.1007/BF00939554.  Google Scholar

[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.  Google Scholar

[1]

Ruotian Gao, Wenxun Xing. Robust sensitivity analysis for linear programming with ellipsoidal perturbation. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-16. doi: 10.3934/jimo.2019041
[2]

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
[3]

Qilin Wang, Shengji Li, Kok Lay Teo. Continuity of second-order adjacent derivatives for weak perturbation maps in vector optimization. Numerical Algebra, Control & Optimization, 2011, 1 (3) : 417-433. doi: 10.3934/naco.2011.1.417
[4]

Alireza Ghaffari Hadigheh, Tamás Terlaky. Generalized support set invariancy sensitivity analysis in linear optimization. Journal of Industrial & Management Optimization, 2006, 2 (1) : 1-18. doi: 10.3934/jimo.2006.2.1
[5]

Behrouz Kheirfam, Kamal mirnia. Comments on ''Generalized support set invariancy sensitivity analysis in linear optimization''. Journal of Industrial & Management Optimization, 2008, 4 (3) : 611-616. doi: 10.3934/jimo.2008.4.611
[6]

Lluís Alsedà, David Juher, Pere Mumbrú. Minimal dynamics for tree maps. Discrete & Continuous Dynamical Systems - A, 2008, 20 (3) : 511-541. doi: 10.3934/dcds.2008.20.511
[7]

Angela Cadena, Adriana Marcucci, Juan F. Pérez, Hernando Durán, Hernando Mutis, Camilo Taútiva, Fernando Palacios. Efficiency analysis in electricity transmission utilities. Journal of Industrial & Management Optimization, 2009, 5 (2) : 253-274. doi: 10.3934/jimo.2009.5.253
[8]

Fahd Jarad, Thabet Abdeljawad. Generalized fractional derivatives and Laplace transform. Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : 709-722. doi: 10.3934/dcdss.2020039
[9]

Boju Jiang, Jaume Llibre. Minimal sets of periods for torus maps. Discrete & Continuous Dynamical Systems - A, 1998, 4 (2) : 301-320. doi: 10.3934/dcds.1998.4.301
[10]

Timothy Blass, Rafael de la Llave. Perturbation and numerical methods for computing the minimal average energy. Networks & Heterogeneous Media, 2011, 6 (2) : 241-255. doi: 10.3934/nhm.2011.6.241
[11]

Wu Chanti, Qiu Youzhen. A nonlinear empirical analysis on influence factor of circulation efficiency. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 929-940. doi: 10.3934/dcdss.2019062
[12]

Runqin Hao, Guanwen Zhang, Dong Li, Jie Zhang. Data modeling analysis on removal efficiency of hexavalent chromium. Mathematical Foundations of Computing, 2019, 2 (3) : 203-213. doi: 10.3934/mfc.2019014
[13]

Fahd Jarad, Yassine Adjabi, Thabet Abdeljawad, Saed F. Mallak, Hussam Alrabaiah. Lyapunov type inequality in the frame of generalized Caputo derivatives. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 0-0. doi: 10.3934/dcdss.2020212
[14]

Fahd Jarad, Thabet Abdeljawad. Variational principles in the frame of certain generalized fractional derivatives. Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : 695-708. doi: 10.3934/dcdss.2020038
[15]

Xin Zuo, Chun-Rong Chen, Hong-Zhi Wei. Solution continuity of parametric generalized vector equilibrium problems with strictly pseudomonotone mappings. Journal of Industrial & Management Optimization, 2017, 13 (1) : 477-488. doi: 10.3934/jimo.2016027
[16]

Xiao-Wen Chang, Ren-Cang Li. Multiplicative perturbation analysis for QR factorizations. Numerical Algebra, Control & Optimization, 2011, 1 (2) : 301-316. doi: 10.3934/naco.2011.1.301
[17]

Ramzi Alsaedi. Perturbation effects for the minimal surface equation with multiple variable exponents. Discrete & Continuous Dynamical Systems - S, 2019, 12 (2) : 139-150. doi: 10.3934/dcdss.2019010
[18]

Annie Millet, Svetlana Roudenko. Generalized KdV equation subject to a stochastic perturbation. Discrete & Continuous Dynamical Systems - B, 2018, 23 (3) : 1177-1198. doi: 10.3934/dcdsb.2018147
[19]

Deren Han, Xiaoming Yuan. Existence of anonymous link tolls for decentralizing an oligopolistic game and the efficiency analysis. Journal of Industrial & Management Optimization, 2011, 7 (2) : 347-364. doi: 10.3934/jimo.2011.7.347
[20]

Cheng-Kai Hu, Fung-Bao Liu, Cheng-Feng Hu. Efficiency measures in fuzzy data envelopment analysis with common weights. Journal of Industrial & Management Optimization, 2017, 13 (1) : 237-249. doi: 10.3934/jimo.2016014

2018 Impact Factor: 1.048

Tools

Metrics

  • PDF downloads (27)
  • HTML views (36)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences