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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 & 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.   Google Scholar

[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).   Google Scholar

[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.  doi: 10.1016/j.jmaa.2004.03.014.  Google Scholar

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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.  doi: 10.1007/s10957-007-9336-4.  Google Scholar

[5]

Y. F. Chai, Y. J. Cho and J. Li, Some characterizations of ideal point in vector optimization problems,, J. Inequal. Appl., 2008 (2008).   Google Scholar

[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.  doi: 10.3934/jimo.2011.7.157.  Google Scholar

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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.  doi: 10.1016/j.camwa.2010.08.036.  Google Scholar

[8]

G. Y. Chen, X. X. Huang and X. Q. Yang, "Vector Optimization: Set-Valued and Variational Analysis,", in, 541 (2005).   Google Scholar

[9]

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

[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.   Google Scholar

[11]

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

[12]

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

[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.   Google Scholar

[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.   Google Scholar

[15]

F. Giannessi, Theorems of alterative, quadratic programs and complementarity problems,, in, (1980), 151.   Google Scholar

[16]

F. Giannessi, "Vector Variational Inequalities and Vector Equilibria: Mathematical Theories,", Kluwer Academic Publishers, (2000).   Google Scholar

[17]

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

[18]

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

[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.  doi: 10.1007/s10898-004-2694-7.  Google Scholar

[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.  doi: 10.3934/jimo.2008.4.167.  Google Scholar

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

[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.  doi: 10.1023/A:1014830925232.  Google Scholar

[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.  doi: 10.3934/jimo.2012.8.411.  Google Scholar

[25]

R. T. Rochafellar, "Convex Analysis,", Princeton University Press, (1970).   Google Scholar

[26]

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

[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.  doi: 10.1023/A:1020844423345.  Google Scholar

[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.  doi: 10.3934/jimo.2012.8.485.  Google Scholar

[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.  doi: 10.3934/jimo.2010.6.895.  Google Scholar

[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.  doi: 10.1007/s10957-010-9732-z.  Google Scholar

[31]

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

show all references

References:
[1]

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

[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).   Google Scholar

[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.  doi: 10.1016/j.jmaa.2004.03.014.  Google Scholar

[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.  doi: 10.1007/s10957-007-9336-4.  Google Scholar

[5]

Y. F. Chai, Y. J. Cho and J. Li, Some characterizations of ideal point in vector optimization problems,, J. Inequal. Appl., 2008 (2008).   Google Scholar

[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.  doi: 10.3934/jimo.2011.7.157.  Google Scholar

[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.  doi: 10.1016/j.camwa.2010.08.036.  Google Scholar

[8]

G. Y. Chen, X. X. Huang and X. Q. Yang, "Vector Optimization: Set-Valued and Variational Analysis,", in, 541 (2005).   Google Scholar

[9]

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

[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.   Google Scholar

[11]

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

[12]

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

[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.   Google Scholar

[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.   Google Scholar

[15]

F. Giannessi, Theorems of alterative, quadratic programs and complementarity problems,, in, (1980), 151.   Google Scholar

[16]

F. Giannessi, "Vector Variational Inequalities and Vector Equilibria: Mathematical Theories,", Kluwer Academic Publishers, (2000).   Google Scholar

[17]

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

[18]

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

[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.  doi: 10.1007/s10898-004-2694-7.  Google Scholar

[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.  doi: 10.3934/jimo.2008.4.167.  Google Scholar

[21]

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

[22]

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

[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.  doi: 10.1023/A:1014830925232.  Google Scholar

[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.  doi: 10.3934/jimo.2012.8.411.  Google Scholar

[25]

R. T. Rochafellar, "Convex Analysis,", Princeton University Press, (1970).   Google Scholar

[26]

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

[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.  doi: 10.1023/A:1020844423345.  Google Scholar

[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.  doi: 10.3934/jimo.2012.8.485.  Google Scholar

[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.  doi: 10.3934/jimo.2010.6.895.  Google Scholar

[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.  doi: 10.1007/s10957-010-9732-z.  Google Scholar

[31]

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

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