American Institute of Mathematical Sciences

Advanced
2013, 3(4): 643-653. doi: 10.3934/naco.2013.3.643

Characterizations of the $E$-Benson proper efficiency in vector optimization problems

Kequan Zhao 1, and  Xinmin Yang 1,

1. 

College of Mathematics Science, Chongqing Normal University, Chongqing 401331, China, China

Received  August 2013 Revised  October 2013 Published  October 2013

In this paper, under the nearly $E$-subconvexlikeness, some characterizations of the $E$-Benson proper efficiency are established in terms of scalarization, Lagrange multipliers, saddle point criteria and duality for a vector optimization problem with set-valued maps. Our main results generalize and unify some previously known results.
Keywords: duality., Vector optimization, saddle point criteria, scalarization, $E$-Benson proper efficiency, Lagrange multipliers, nearly $E$-Subconvexlike-ness.
Mathematics Subject Classification: Primary: 49N15; Secondary: 90C26, 90C29, 90C4.
Citation: Kequan Zhao, Xinmin Yang. Characterizations of the $E$-Benson proper efficiency in vector optimization problems. Numerical Algebra, Control & Optimization, 2013, 3 (4) : 643-653. doi: 10.3934/naco.2013.3.643
References:
[1]

H. P. Benson, An improved definition of proper efficiency for vector maximization with respect to cones,, J. Math. Anal. Appl., 71 (1979), 232. doi: 10.1016/0022-247X(79)90226-9. Google Scholar

[2]

J. Borwein, Proper efficient points for maximizations with respect to cones,, SIAM. J. Control and Optim., 15 (1977), 57. Google Scholar

[3]

G. Y. Chen and W. D. Rong, Characterizations of the Benson proper efficiency for nonconvex vector optimization,, J. Optim. Theory Appl., 98 (1998), 365. doi: 10.1023/A:1022689517921. Google Scholar

[4]

G. Y. Chen, X. X. Huang and X. Q. Yang, "Vector Optimization. Lecture Notes in Economics and Mathematical Sciences, 541,", Springer, (2005). Google Scholar

[5]

M. Chicco, F. Mignanego, L. Pusillo and S. Tijs, Vector optimization problems via improvement sets,, J. Optim. Theory Appl., 150 (2011), 516. doi: 10.1007/s10957-011-9851-1. Google Scholar

[6]

M. Ehrgott, "Multicriteria Optimization,", Springer, (2005). Google Scholar

[7]

Y. Gao and X. M. Yang, Optimality conditions for approximate solutions of vector optimization problems,, J. Ind. Manag. Optim., 7 (2011), 483. doi: 10.3934/jimo.2011.7.483. Google Scholar

[8]

A. M. Geffrion, Proper efficiency and the theory of vector maximization,, J. Math. Anal. Appl., 22 (1968), 618. Google Scholar

[9]

B. A. Ghaznavi-ghosoni, E. Khorram and M. Soleimani-damaneh, Scalarization for characterization of approximate strong/weak/proper efficiency in multiobjective optimization,, Optimization, 62 (2013), 703. doi: 10.1080/02331934.2012.668190. Google Scholar

[10]

C. Gutiérrez, B. Jiménez and V. Novo, Improvement sets and vector optimization,, Eur. J. Oper. Res., 223 (2012), 304. Google Scholar

[11]

C. Gutiérrez, B. Jiménez and V. Novo, A unified approach and optimality conditions for approximate solutions of vector optimization problems,, SIAM J. Optim., 17 (2006), 688. Google Scholar

[12]

C. Gutiérrez, L. Huerga and V. Novo, Scalarization and saddle points of approximate proper solutions in nearly subconvexlike vector optimization problems,, J. Math. Anal. Appl., 389 (2012), 1046. doi: 10.1016/j.jmaa.2011.12.050. Google Scholar

[13]

M. I. Henig, Proper efficiency with respect to cones,, J. Optim. Theory Appl., 36 (1982), 387. doi: 10.1007/BF00934353. Google Scholar

[14]

J. Jahn, "Vector Optimization. Theory, Applications, and Extensions,", Springer, (2004). Google Scholar

[15]

Z. F. Li, Benson proper efficiency in the vector optimization of set-valued maps,, J. Optim. Theory Appl., 98 (1998), 623. doi: 10.1023/A:1022676013609. Google Scholar

[16]

J. C. Liu, ε-Properly efficient solutions to nondifferentiable multiobjective programming problems,, Appl. Math. Lett., 12 (1999), 109. doi: 10.1016/S0893-9659(99)00087-7. Google Scholar

[17]

D. T. Luc, "Theory of Vector Optimization. Lecture Notes in Economics and Mathematical Sciences, 319,", Springer, (1988). Google Scholar

[18]

W. D. Rong and Y. Ma, ε-Properly efficient solutions of vector optimization problems with set-valued maps,, OR Transactions, 4 (2000), 21. Google Scholar

[19]

X. M. Yang, D. Li and S. Y. Wang, Near-subconvexlikeness in vector optimization with set-valued functions,, J. Optim. Theory Appl., 110 (2001), 413. doi: 10.1023/A:1017535631418. Google Scholar

[20]

X. M. Yang, X. Q. Yang and G. Y. Chen, Theorems of the alternative and optimization with set-valued maps,, J. Optim. Theory Appl., 107 (2000), 627. doi: 10.1023/A:1004613630675. Google Scholar

[21]

K. Q. Zhao and X. M. Yang, E-Benson proper efficiency in vector optimization,, Optimization, (2013). doi: 10.1080/02331934.2013.798321. Google Scholar

[22]

K. Q. Zhao, X. M. Yang and J. W. Peng, Weak E-Optimal solution in vector optimization,, Taiwan. J. Math., 17 (2013), 1287. Google Scholar

show all references

References:
[1]

H. P. Benson, An improved definition of proper efficiency for vector maximization with respect to cones,, J. Math. Anal. Appl., 71 (1979), 232. doi: 10.1016/0022-247X(79)90226-9. Google Scholar

[2]

J. Borwein, Proper efficient points for maximizations with respect to cones,, SIAM. J. Control and Optim., 15 (1977), 57. Google Scholar

[3]

G. Y. Chen and W. D. Rong, Characterizations of the Benson proper efficiency for nonconvex vector optimization,, J. Optim. Theory Appl., 98 (1998), 365. doi: 10.1023/A:1022689517921. Google Scholar

[4]

G. Y. Chen, X. X. Huang and X. Q. Yang, "Vector Optimization. Lecture Notes in Economics and Mathematical Sciences, 541,", Springer, (2005). Google Scholar

[5]

M. Chicco, F. Mignanego, L. Pusillo and S. Tijs, Vector optimization problems via improvement sets,, J. Optim. Theory Appl., 150 (2011), 516. doi: 10.1007/s10957-011-9851-1. Google Scholar

[6]

M. Ehrgott, "Multicriteria Optimization,", Springer, (2005). Google Scholar

[7]

Y. Gao and X. M. Yang, Optimality conditions for approximate solutions of vector optimization problems,, J. Ind. Manag. Optim., 7 (2011), 483. doi: 10.3934/jimo.2011.7.483. Google Scholar

[8]

A. M. Geffrion, Proper efficiency and the theory of vector maximization,, J. Math. Anal. Appl., 22 (1968), 618. Google Scholar

[9]

B. A. Ghaznavi-ghosoni, E. Khorram and M. Soleimani-damaneh, Scalarization for characterization of approximate strong/weak/proper efficiency in multiobjective optimization,, Optimization, 62 (2013), 703. doi: 10.1080/02331934.2012.668190. Google Scholar

[10]

C. Gutiérrez, B. Jiménez and V. Novo, Improvement sets and vector optimization,, Eur. J. Oper. Res., 223 (2012), 304. Google Scholar

[11]

C. Gutiérrez, B. Jiménez and V. Novo, A unified approach and optimality conditions for approximate solutions of vector optimization problems,, SIAM J. Optim., 17 (2006), 688. Google Scholar

[12]

C. Gutiérrez, L. Huerga and V. Novo, Scalarization and saddle points of approximate proper solutions in nearly subconvexlike vector optimization problems,, J. Math. Anal. Appl., 389 (2012), 1046. doi: 10.1016/j.jmaa.2011.12.050. Google Scholar

[13]

M. I. Henig, Proper efficiency with respect to cones,, J. Optim. Theory Appl., 36 (1982), 387. doi: 10.1007/BF00934353. Google Scholar

[14]

J. Jahn, "Vector Optimization. Theory, Applications, and Extensions,", Springer, (2004). Google Scholar

[15]

Z. F. Li, Benson proper efficiency in the vector optimization of set-valued maps,, J. Optim. Theory Appl., 98 (1998), 623. doi: 10.1023/A:1022676013609. Google Scholar

[16]

J. C. Liu, ε-Properly efficient solutions to nondifferentiable multiobjective programming problems,, Appl. Math. Lett., 12 (1999), 109. doi: 10.1016/S0893-9659(99)00087-7. Google Scholar

[17]

D. T. Luc, "Theory of Vector Optimization. Lecture Notes in Economics and Mathematical Sciences, 319,", Springer, (1988). Google Scholar

[18]

W. D. Rong and Y. Ma, ε-Properly efficient solutions of vector optimization problems with set-valued maps,, OR Transactions, 4 (2000), 21. Google Scholar

[19]

X. M. Yang, D. Li and S. Y. Wang, Near-subconvexlikeness in vector optimization with set-valued functions,, J. Optim. Theory Appl., 110 (2001), 413. doi: 10.1023/A:1017535631418. Google Scholar

[20]

X. M. Yang, X. Q. Yang and G. Y. Chen, Theorems of the alternative and optimization with set-valued maps,, J. Optim. Theory Appl., 107 (2000), 627. doi: 10.1023/A:1004613630675. Google Scholar

[21]

K. Q. Zhao and X. M. Yang, E-Benson proper efficiency in vector optimization,, Optimization, (2013). doi: 10.1080/02331934.2013.798321. Google Scholar

[22]

K. Q. Zhao, X. M. Yang and J. W. Peng, Weak E-Optimal solution in vector optimization,, Taiwan. J. Math., 17 (2013), 1287. Google Scholar

[1]

Marius Durea, Elena-Andreea Florea, Radu Strugariu. Henig proper efficiency in vector optimization with variable ordering structure. Journal of Industrial & Management Optimization, 2019, 15 (2) : 791-815. doi: 10.3934/jimo.2018071
[2]

Zhiang Zhou, Xinmin Yang, Kequan Zhao. $E$-super efficiency of set-valued optimization problems involving improvement sets. Journal of Industrial & Management Optimization, 2016, 12 (3) : 1031-1039. doi: 10.3934/jimo.2016.12.1031
[3]

Guolin Yu. Global proper efficiency and vector optimization with cone-arcwise connected set-valued maps. Numerical Algebra, Control & Optimization, 2016, 6 (1) : 35-44. doi: 10.3934/naco.2016.6.35
[4]

Ying Gao, Xinmin Yang, Jin Yang, Hong Yan. Scalarizations and Lagrange multipliers for approximate solutions in the vector optimization problems with set-valued maps. Journal of Industrial & Management Optimization, 2015, 11 (2) : 673-683. doi: 10.3934/jimo.2015.11.673
[5]

Tadeusz Antczak, Najeeb Abdulaleem. Optimality conditions for $ E $-differentiable vector optimization problems with the multiple interval-valued objective function. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-19. doi: 10.3934/jimo.2019089
[6]

Caiping Liu, Heungwing Lee. Lagrange multiplier rules for approximate solutions in vector optimization. Journal of Industrial & Management Optimization, 2012, 8 (3) : 749-764. doi: 10.3934/jimo.2012.8.749
[7]

Xinmin Yang. On symmetric and self duality in vector optimization problem. Journal of Industrial & Management Optimization, 2011, 7 (3) : 523-529. doi: 10.3934/jimo.2011.7.523
[8]

Karla L. Cortez, Javier F. Rosenblueth. Normality and uniqueness of Lagrange multipliers. Discrete & Continuous Dynamical Systems - A, 2018, 38 (6) : 3169-3188. doi: 10.3934/dcds.2018138
[9]

Radu Ioan Boţ, Sorin-Mihai Grad. On linear vector optimization duality in infinite-dimensional spaces. Numerical Algebra, Control & Optimization, 2011, 1 (3) : 407-415. doi: 10.3934/naco.2011.1.407
[10]

Jiawei Chen, Shengjie Li, Jen-Chih Yao. Vector-valued separation functions and constrained vector optimization problems: optimality and saddle points. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-18. doi: 10.3934/jimo.2018174
[11]

Qiusheng Qiu, Xinmin Yang. Scalarization of approximate solution for vector equilibrium problems. Journal of Industrial & Management Optimization, 2013, 9 (1) : 143-151. doi: 10.3934/jimo.2013.9.143
[12]

Takeshi Fukao, Nobuyuki Kenmochi. Abstract theory of variational inequalities and Lagrange multipliers. Conference Publications, 2013, 2013 (special) : 237-246. doi: 10.3934/proc.2013.2013.237
[13]

Baoxiang Wang. E-Besov spaces and dissipative equations. Communications on Pure & Applied Analysis, 2004, 3 (4) : 883-919. doi: 10.3934/cpaa.2004.3.883
[14]

Annamaria Barbagallo, Rosalba Di Vincenzo, Stéphane Pia. On strong Lagrange duality for weighted traffic equilibrium problem. Discrete & Continuous Dynamical Systems - A, 2011, 31 (4) : 1097-1113. doi: 10.3934/dcds.2011.31.1097
[15]

Regina Sandra Burachik, Alex Rubinov. On the absence of duality gap for Lagrange-type functions. Journal of Industrial & Management Optimization, 2005, 1 (1) : 33-38. doi: 10.3934/jimo.2005.1.33
[16]

Tao Chen, Yunping Jiang, Gaofei Zhang. No invariant line fields on escaping sets of the family $\lambda e^{iz}+\gamma e^{-iz}$. Discrete & Continuous Dynamical Systems - A, 2013, 33 (5) : 1883-1890. doi: 10.3934/dcds.2013.33.1883
[17]

Lijia Yan. Some properties of a class of $(F,E)$-$G$ generalized convex functions. Numerical Algebra, Control & Optimization, 2013, 3 (4) : 615-625. doi: 10.3934/naco.2013.3.615
[18]

Augusto Visintin. P.D.E.s with hysteresis 30 years later. Discrete & Continuous Dynamical Systems - S, 2015, 8 (4) : 793-816. doi: 10.3934/dcdss.2015.8.793
[19]

Alejandro Cataldo, Juan-Carlos Ferrer, Pablo A. Rey, Antoine Sauré. Design of a single window system for e-government services: the chilean case. Journal of Industrial & Management Optimization, 2018, 14 (2) : 561-582. doi: 10.3934/jimo.2017060
[20]

M. Guru Prem Prasad, Tarakanta Nayak. Dynamics of { $\lambda tanh(e^z): \lambda \in R$\ ${ 0 }$ }. Discrete & Continuous Dynamical Systems - A, 2007, 19 (1) : 121-138. doi: 10.3934/dcds.2007.19.121

 Impact Factor: 

Tools

Metrics

  • PDF downloads (5)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences