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

Shasha Hu; Yihong Xu; Yuhan Zhang

doi:10.3934/jimo.2020164

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2022, Volume 18, Issue 1: 469-486. Doi: 10.3934/jimo.2020164

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

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

^* Corresponding author: Yihong Xu
^* Corresponding author: Yihong Xu

Received: August 31, 2019

Revised: June 30, 2020

Early access: November 2020

Published: January 2022

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

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Abstract

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.

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Mathematics Subject Classification: Primary: 90C33, 90C46; Secondary: 90C59.

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References

[1]	J.-P. Aubin and H. Frankowska, Set-Valued Analysis, Systems & Control: Foundations & Applications, 2. Birkhäuser Boston, Inc., Boston, MA, 1990.
[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.
[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.
[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.
[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.
[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.
[7]	J. Jahn, Vector Optimization: Theory, Applications, and Extensions, Springer-Verlag, Berlin, 2004. doi: 10.1007/978-3-540-24828-6.
[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.
[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.
[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.
[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.
[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.
[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.
[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.