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January  2022, 18(1): 469-486. doi: 10.3934/jimo.2020164

Second-Order characterizations for set-valued equilibrium problems with variable ordering structures

Shasha Hu , Yihong Xu , and  Yuhan Zhang

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

* Corresponding author: Yihong Xu

Received  September 2019 Revised  July 2020 Published  January 2022 Early access  November 2020

Fund Project: This research was supported by the National Natural Science Foundation of China Grant (11961047) and the Natural Science Foundation of Jiangxi Province (20192BAB201010)

The concepts of weakly efficient solutions and globally efficient solutions are introduced for constrained set-valued equilibrium problems with variable ordering structures. By applying the second-order tangent epiderivative and a nonlinear functional, necessary optimality conditions for weakly efficient solutions and globally efficient solutions are established without any convexity assumption. Under the cone-convexity of the objective and constraint functions, sufficient optimality conditions are given. In addition, the tangent derivatives of objective and constraint functions are separated. Simultaneously, a unified necessary and sufficient optimality conditions for weakly efficient solutions is derived, and the same goes for globally efficient solutions. In particular, we give specific examples to illustrate the optimality conditions, respectively.

Keywords: Set-valued equilibrium problem, variable ordering structure, second-order tangent derivative, nonlinear scalarzation, optimality condition.
Mathematics Subject Classification: Primary: 90C33, 90C46; Secondary: 90C59.
Citation: Shasha Hu, Yihong Xu, Yuhan Zhang. Second-Order characterizations for set-valued equilibrium problems with variable ordering structures. Journal of Industrial & Management Optimization, 2022, 18 (1) : 469-486. doi: 10.3934/jimo.2020164
References:
[1]

J.-P. Aubin and H. Frankowska, Set-Valued Analysis, Systems & Control: Foundations & Applications, 2. Birkhäuser Boston, Inc., Boston, MA, 1990.  Google Scholar

[2]

G. Y. Chen, Existence of solutions for a vector variational inequality: An extension of the Hartmann-Stampacchia theorem, J. Optim. Theory Appl., 74 (1992), 445-456.  doi: 10.1007/BF00940320.  Google Scholar

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M. Durea, R. Strugariu and C. Tammer, On set-valued optimization problems with variable ordering structure, J. Glob. Optim., 61 (2015), 745-767.  doi: 10.1007/s10898-014-0207-x.  Google Scholar

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G. Eichfelder, Variable ordering structures in vector optimization, Recent Developments in Vector Optimization, Vector Optim., Springer, Berlin, (2012), 95–126. doi: 10.1007/978-3-642-21114-0_4.  Google Scholar

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

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P. Q. Khanh and N. M. Tung, Second-order conditions for open-cone minimizers and firm minimizers in set-valued optimization subject to mixed constraints, J. Optim. Theory Appl., 171 (2016), 45-69.  doi: 10.1007/s10957-016-0995-x.  Google Scholar

[9]

Z. H. Peng and Y. H. Xu, New second-order tangent epiderivatives and applications to set-valued optimization, J. Optim. Theory Appl., 172 (2017), 128-140.  doi: 10.1007/s10957-016-1011-1.  Google Scholar

[10]

Q. S. Qiu and X. M. Yang, Some properties of approximate solutions for vector optimization problem with set-valued functions, J. Glob. Optim., 47 (2010), 1-12.  doi: 10.1007/s10898-009-9452-9.  Google Scholar

[11]

B. Soleimani, Characterization of approximate solutions of vector optimization problems with variable order structure, J. Optim. Theory Appl., 162 (2014), 605-632.  doi: 10.1007/s10957-014-0535-5.  Google Scholar

[12]

C. Tammer and P. Weidner, Nonconvex separation theorem and some applications in vector optimization, J. Optim. Theory Appl., 67 (1990), 297-320.  doi: 10.1007/BF00940478.  Google Scholar

[13] P. L. Yu, Multiple-Criteria Decision Making: Concepts, Techniques, and Extensions, Mathematical Concepts and Methods in Science and Engineering, 30. Plenum Press, New York, 1985.  doi: 10.1007/978-1-4684-8395-6.  Google Scholar
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S. K. Zhu, S. J. Li and K. L. Teo, Second-order Karush-Kuhn-Tucker optimality conditions for set-valued optimization, J. Glob. Optim., 58 (2014), 673-692.  doi: 10.1007/s10898-013-0067-9.  Google Scholar

show all references

References:
[1]

J.-P. Aubin and H. Frankowska, Set-Valued Analysis, Systems & Control: Foundations & Applications, 2. Birkhäuser Boston, Inc., Boston, MA, 1990.  Google Scholar

[2]

G. Y. Chen, Existence of solutions for a vector variational inequality: An extension of the Hartmann-Stampacchia theorem, J. Optim. Theory Appl., 74 (1992), 445-456.  doi: 10.1007/BF00940320.  Google Scholar

[3]

G. Y. Chen and X. Q. Yang, Characterizations of variable domination structures via nonlinear scalarization, J. Optim. Theory Appl., 112 (2002), 97-110.  doi: 10.1023/A:1013044529035.  Google Scholar

[4]

M. Durea, R. Strugariu and C. Tammer, On set-valued optimization problems with variable ordering structure, J. Glob. Optim., 61 (2015), 745-767.  doi: 10.1007/s10898-014-0207-x.  Google Scholar

[5]

G. Eichfelder, Variable ordering structures in vector optimization, Recent Developments in Vector Optimization, Vector Optim., Springer, Berlin, (2012), 95–126. doi: 10.1007/978-3-642-21114-0_4.  Google Scholar

[6]

X.-H. Gong, Scalarization and optimality conditions for vector equilibrium problems, Nonlinear Anal. TMA, 73 (2010), 3598-3612.  doi: 10.1016/j.na.2010.07.041.  Google Scholar

[7]

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

[8]

P. Q. Khanh and N. M. Tung, Second-order conditions for open-cone minimizers and firm minimizers in set-valued optimization subject to mixed constraints, J. Optim. Theory Appl., 171 (2016), 45-69.  doi: 10.1007/s10957-016-0995-x.  Google Scholar

[9]

Z. H. Peng and Y. H. Xu, New second-order tangent epiderivatives and applications to set-valued optimization, J. Optim. Theory Appl., 172 (2017), 128-140.  doi: 10.1007/s10957-016-1011-1.  Google Scholar

[10]

Q. S. Qiu and X. M. Yang, Some properties of approximate solutions for vector optimization problem with set-valued functions, J. Glob. Optim., 47 (2010), 1-12.  doi: 10.1007/s10898-009-9452-9.  Google Scholar

[11]

B. Soleimani, Characterization of approximate solutions of vector optimization problems with variable order structure, J. Optim. Theory Appl., 162 (2014), 605-632.  doi: 10.1007/s10957-014-0535-5.  Google Scholar

[12]

C. Tammer and P. Weidner, Nonconvex separation theorem and some applications in vector optimization, J. Optim. Theory Appl., 67 (1990), 297-320.  doi: 10.1007/BF00940478.  Google Scholar

[13] P. L. Yu, Multiple-Criteria Decision Making: Concepts, Techniques, and Extensions, Mathematical Concepts and Methods in Science and Engineering, 30. Plenum Press, New York, 1985.  doi: 10.1007/978-1-4684-8395-6.  Google Scholar
[14]

S. K. Zhu, S. J. Li and K. L. Teo, Second-order Karush-Kuhn-Tucker optimality conditions for set-valued optimization, J. Glob. Optim., 58 (2014), 673-692.  doi: 10.1007/s10898-013-0067-9.  Google Scholar

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