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January  2013, 9(1): 57-74. doi: 10.3934/jimo.2013.9.57

Stability analysis for set-valued vector mixed variational inequalities in real reflexive Banach spaces

Xing Wang 1, and  Nan-Jing Huang 2,

1. 

Department of Mathematics, Sichuan University, Chengdu, Sichuan, 610064, China
2. 

Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064

Received  November 2011 Revised  May 2012 Published  December 2012

In this paper, some characterizations for the solution sets of a class of set-valued vector mixed variational inequalities to be nonempty and bounded are presented in real reflexive Banach spaces. An equivalence relation between the solution sets of the vector mixed variational inequalities and the weakly efficient solution sets of the vector optimization problems is shown under some suitable assumptions. By using some known results for the vector optimization problems, several characterizations for the solution sets of the vector mixed variational inequalities are obtained in real reflexive Banach spaces. Furthermore, some stability results for the vector mixed variational inequality are given when the mapping and the constraint set are perturbed by two different parameters. Finally, the upper semicontinuity and the lower semicontinuity of the solution sets are given under some suitable assumptions which are different from the ones used in [7, 11, 22]. Some examples are also given to illustrate our results.
Keywords: stability, vector optimization problem, upper semicontinuity, boundedness, nonemptiness, Vector mixed variational inequality, lower semicontinuity..
Mathematics Subject Classification: 49J4.
Citation: Xing Wang, Nan-Jing Huang. Stability analysis for set-valued vector mixed variational inequalities in real reflexive Banach spaces. Journal of Industrial and Management Optimization, 2013, 9 (1) : 57-74. doi: 10.3934/jimo.2013.9.57
References:
[1]

M. Adivar and S. C. Fang, Convex optimization on mixed domains, J. Ind. Manag. Optim., 8 (2012), 189-227.

[2]

R. P. Agarwal, Y. J. Cho and N. Petrot, Systems of general nonlinear set-valued mixed variational inequalities problems in Hilbert spaces, Fixed Point Theory Appl., 2011 (2011), pp10.

[3]

L. Q. Anh and P. Q. Khanh, Semicontinuity of the solution set of parametric multivalued vector quasiequilibrium problems, J. Math. Anal. Appl., 294 (2004), 699-711. doi: 10.1016/j.jmaa.2004.03.014.

[4]

L. C. Ceng, S. Schaible and J. C. Yao, Existence of solutions for generalized vector variational-like inequalities, J. Optim. Theory Appl., 137 (2008), 121-133. doi: 10.1007/s10957-007-9336-4.

[5]

Y. F. Chai, Y. J. Cho and J. Li, Some characterizations of ideal point in vector optimization problems, J. Inequal. Appl., 2008 (2008), pp.8. Art. ID 231845.

[6]

C. R. Chen, T. C. Edwin Cheng, S. J. Li and X. Q. Yang, Nolinear augmented lagrangian for nonconvex multiobjective optimization, J. Ind. Manag. Optim., 7 (2011), 157-174. doi: 10.3934/jimo.2011.7.157.

[7]

C. R. Chen, S. J. Li and Z. M. Fang, On the solution semicontinuity to a parametric generalized vector quasivariational inequality, Comput. Math. Appl., 60 (2010), 2417-2425. doi: 10.1016/j.camwa.2010.08.036.

[8]

G. Y. Chen, X. X. Huang and X. Q. Yang, "Vector Optimization: Set-Valued and Variational Analysis," in "Lecture Notes in Economics and Mathematical System" 541, Springer, Berlin, 2005.

[9]

G. Y. Chen and S. J. Li, Existence of solutions for a generalized vector quasivariational inequality, J. Optim. Theory Appl., 90 (1996), 321-334. doi: 10.1007/BF02190001.

[10]

J. W. Chen, Y. J. Cho, J. K. Kim and J. Li, Multiobjective optimization problems with modified objective functions and cone constraints and applications, J. Global Optim., 49 (2011), 137-147.

[11]

Y. H. Cheng and D. L. Zhu, Global stability results for the weak vector variational inequality, J. Global Optim., 32 (2005), 543-550.

[12]

Y. Chiang, Semicontinuous mappings into T. V. S. with applications to mixed vector variational-like inequalities, J. Global Optim., 32 (2005), 467-484. doi: 10.1007/s10898-003-2684-1.

[13]

Y. J. Cho and N. Petrot, An optimization problem related to generalized equilibrium and fixed point problems with applications, Fixed Point Theory, 11 (2010), 237-250.

[14]

S. Deng, Characterizations of the nonemptiness and boundedness of weakly efficient solution sets of convex vector optimization problems in real reflexive Banach spaces, J. Optim. Theory Appl., 140 (2009), 1-7.

[15]

F. Giannessi, Theorems of alterative, quadratic programs and complementarity problems, in "Variational Inequalities and Complementarity Problems" (eds. R. W. Cottle, F. Giannessi and J. L. Lions), Wiley &Sons, New York, (1980), 151-186.

[16]

F. Giannessi, "Vector Variational Inequalities and Vector Equilibria: Mathematical Theories," Kluwer Academic Publishers, Dordrecht, Holland, 2000.

[17]

Y. R. He, Stable pseudomonotone variational inequality in reflexive Banach space, J. Math. Anal. Appl., 330 (2007), 352-363.

[18]

N. J. Huang and Y. P. Fang, On vector variational inequalities in reflexive Banach spaces, J. Global Optim., 32 (2005), 495-505.

[19]

P. Q. Khanh and L. M. Luu, Upper semicontinuity of the solution set to parametric vector quasivariational inequalities, J. Global Optim., 32 (2005), 569-580. doi: 10.1007/s10898-004-2694-7.

[20]

K. Kimura and J. C. Yao, Semicontinuity of solution mappings of parametric generalized strong vector equilibrium problems, J. Ind. Manag. Optim., 4 (2008), 167-181. doi: 10.3934/jimo.2008.4.167.

[21]

G. M. Lee and K. B. Lee, Vector variational inequalities for nondifferentiable convex vector optimization problems, J. Glob. Optim., 32 (2005), 597-612. doi: 10.1007/s10898-004-2696-5.

[22]

S. J. Li and C. R. Chen, Stability of weak vector variational inequality, Nonlinear Anal., 70 (2009), 1528-1535.

[23]

S. J. Li, G. Y. Chen and K. L. Teo, On the stability of generalized vector quasivariational inequality problems, J. Optim. Theory Appl., 113 (2002), 283-295. doi: 10.1023/A:1014830925232.

[24]

S. J. Li and X. L. Guo, Calculus rules of generalized $\varepsilon$-subdifferential for vector valued mappings and applications, J. Ind. Manag. Optim., 8 (2012), 411-427. doi: 10.3934/jimo.2012.8.411.

[25]

R. T. Rochafellar, "Convex Analysis," Princeton University Press, Princeton, 1970.

[26]

M. Sion, On general minimax theorems, Pacific J. Math., 8 (1958), 171-176.

[27]

X. Q. Yang and J. C. Yao, Gap functions and existence of solutions to set-valued vector variational inequalities, J. Optim. Theory Appl., 115 (2002), 407-417. doi: 10.1023/A:1020844423345.

[28]

C. J. Yu, K. L. Teo, L. S. Zhang and Y. Q. Bai, On a refinement of the convergence analysis for the new exact penalty function method for continuous inequality constrained optimization problem, J. Ind. Manag. Optim., 8 (2012), 485-491. doi: 10.3934/jimo.2012.8.485.

[29]

C. J. Yu, K. L. Teo, L. S. Zhang and Y. Q. Bai, A new exact penalty function method for continuous inequality constrained optimization problems, J. Ind. Manag. Optim., 6 (2010), 895-910. doi: 10.3934/jimo.2010.6.895.

[30]

R. Y. Zhong and N. J. Huang, Stability analysis for minty mixed variational inequality in reflexive Banach spaces, J. Optim. Theory Appl., 147 (2010), 454-472. doi: 10.1007/s10957-010-9732-z.

[31]

J. Zhu, Y. J. Zhong and Y. J. Cho, Generalized variational principle and vector optimization, J. Optim. Theory Appl., 106 (2000), 201-217. doi: 10.1023/A:1004619426652.

show all references

References:
[1]

M. Adivar and S. C. Fang, Convex optimization on mixed domains, J. Ind. Manag. Optim., 8 (2012), 189-227.

[2]

R. P. Agarwal, Y. J. Cho and N. Petrot, Systems of general nonlinear set-valued mixed variational inequalities problems in Hilbert spaces, Fixed Point Theory Appl., 2011 (2011), pp10.

[3]

L. Q. Anh and P. Q. Khanh, Semicontinuity of the solution set of parametric multivalued vector quasiequilibrium problems, J. Math. Anal. Appl., 294 (2004), 699-711. doi: 10.1016/j.jmaa.2004.03.014.

[4]

L. C. Ceng, S. Schaible and J. C. Yao, Existence of solutions for generalized vector variational-like inequalities, J. Optim. Theory Appl., 137 (2008), 121-133. doi: 10.1007/s10957-007-9336-4.

[5]

Y. F. Chai, Y. J. Cho and J. Li, Some characterizations of ideal point in vector optimization problems, J. Inequal. Appl., 2008 (2008), pp.8. Art. ID 231845.

[6]

C. R. Chen, T. C. Edwin Cheng, S. J. Li and X. Q. Yang, Nolinear augmented lagrangian for nonconvex multiobjective optimization, J. Ind. Manag. Optim., 7 (2011), 157-174. doi: 10.3934/jimo.2011.7.157.

[7]

C. R. Chen, S. J. Li and Z. M. Fang, On the solution semicontinuity to a parametric generalized vector quasivariational inequality, Comput. Math. Appl., 60 (2010), 2417-2425. doi: 10.1016/j.camwa.2010.08.036.

[8]

G. Y. Chen, X. X. Huang and X. Q. Yang, "Vector Optimization: Set-Valued and Variational Analysis," in "Lecture Notes in Economics and Mathematical System" 541, Springer, Berlin, 2005.

[9]

G. Y. Chen and S. J. Li, Existence of solutions for a generalized vector quasivariational inequality, J. Optim. Theory Appl., 90 (1996), 321-334. doi: 10.1007/BF02190001.

[10]

J. W. Chen, Y. J. Cho, J. K. Kim and J. Li, Multiobjective optimization problems with modified objective functions and cone constraints and applications, J. Global Optim., 49 (2011), 137-147.

[11]

Y. H. Cheng and D. L. Zhu, Global stability results for the weak vector variational inequality, J. Global Optim., 32 (2005), 543-550.

[12]

Y. Chiang, Semicontinuous mappings into T. V. S. with applications to mixed vector variational-like inequalities, J. Global Optim., 32 (2005), 467-484. doi: 10.1007/s10898-003-2684-1.

[13]

Y. J. Cho and N. Petrot, An optimization problem related to generalized equilibrium and fixed point problems with applications, Fixed Point Theory, 11 (2010), 237-250.

[14]

S. Deng, Characterizations of the nonemptiness and boundedness of weakly efficient solution sets of convex vector optimization problems in real reflexive Banach spaces, J. Optim. Theory Appl., 140 (2009), 1-7.

[15]

F. Giannessi, Theorems of alterative, quadratic programs and complementarity problems, in "Variational Inequalities and Complementarity Problems" (eds. R. W. Cottle, F. Giannessi and J. L. Lions), Wiley &Sons, New York, (1980), 151-186.

[16]

F. Giannessi, "Vector Variational Inequalities and Vector Equilibria: Mathematical Theories," Kluwer Academic Publishers, Dordrecht, Holland, 2000.

[17]

Y. R. He, Stable pseudomonotone variational inequality in reflexive Banach space, J. Math. Anal. Appl., 330 (2007), 352-363.

[18]

N. J. Huang and Y. P. Fang, On vector variational inequalities in reflexive Banach spaces, J. Global Optim., 32 (2005), 495-505.

[19]

P. Q. Khanh and L. M. Luu, Upper semicontinuity of the solution set to parametric vector quasivariational inequalities, J. Global Optim., 32 (2005), 569-580. doi: 10.1007/s10898-004-2694-7.

[20]

K. Kimura and J. C. Yao, Semicontinuity of solution mappings of parametric generalized strong vector equilibrium problems, J. Ind. Manag. Optim., 4 (2008), 167-181. doi: 10.3934/jimo.2008.4.167.

[21]

G. M. Lee and K. B. Lee, Vector variational inequalities for nondifferentiable convex vector optimization problems, J. Glob. Optim., 32 (2005), 597-612. doi: 10.1007/s10898-004-2696-5.

[22]

S. J. Li and C. R. Chen, Stability of weak vector variational inequality, Nonlinear Anal., 70 (2009), 1528-1535.

[23]

S. J. Li, G. Y. Chen and K. L. Teo, On the stability of generalized vector quasivariational inequality problems, J. Optim. Theory Appl., 113 (2002), 283-295. doi: 10.1023/A:1014830925232.

[24]

S. J. Li and X. L. Guo, Calculus rules of generalized $\varepsilon$-subdifferential for vector valued mappings and applications, J. Ind. Manag. Optim., 8 (2012), 411-427. doi: 10.3934/jimo.2012.8.411.

[25]

R. T. Rochafellar, "Convex Analysis," Princeton University Press, Princeton, 1970.

[26]

M. Sion, On general minimax theorems, Pacific J. Math., 8 (1958), 171-176.

[27]

X. Q. Yang and J. C. Yao, Gap functions and existence of solutions to set-valued vector variational inequalities, J. Optim. Theory Appl., 115 (2002), 407-417. doi: 10.1023/A:1020844423345.

[28]

C. J. Yu, K. L. Teo, L. S. Zhang and Y. Q. Bai, On a refinement of the convergence analysis for the new exact penalty function method for continuous inequality constrained optimization problem, J. Ind. Manag. Optim., 8 (2012), 485-491. doi: 10.3934/jimo.2012.8.485.

[29]

C. J. Yu, K. L. Teo, L. S. Zhang and Y. Q. Bai, A new exact penalty function method for continuous inequality constrained optimization problems, J. Ind. Manag. Optim., 6 (2010), 895-910. doi: 10.3934/jimo.2010.6.895.

[30]

R. Y. Zhong and N. J. Huang, Stability analysis for minty mixed variational inequality in reflexive Banach spaces, J. Optim. Theory Appl., 147 (2010), 454-472. doi: 10.1007/s10957-010-9732-z.

[31]

J. Zhu, Y. J. Zhong and Y. J. Cho, Generalized variational principle and vector optimization, J. Optim. Theory Appl., 106 (2000), 201-217. doi: 10.1023/A:1004619426652.

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