• Previous Article
    A class of nonlinear Lagrangian algorithms for minimax problems
  • JIMO Home
  • This Issue
  • Next Article
    Production-distribution planning of construction supply chain management under fuzzy random environment for large-scale construction projects
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

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.
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

[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

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

[1]

Ying Lin, Qi Ye. Support vector machine classifiers by non-Euclidean margins. Mathematical Foundations of Computing, 2020, 3 (4) : 279-300. doi: 10.3934/mfc.2020018

[2]

Alberto Bressan, Sondre Tesdal Galtung. A 2-dimensional shape optimization problem for tree branches. Networks & Heterogeneous Media, 2020  doi: 10.3934/nhm.2020031

[3]

Gloria Paoli, Gianpaolo Piscitelli, Rossanno Sannipoli. A stability result for the Steklov Laplacian Eigenvalue Problem with a spherical obstacle. Communications on Pure & Applied Analysis, 2021, 20 (1) : 145-158. doi: 10.3934/cpaa.2020261

[4]

Lingfeng Li, Shousheng Luo, Xue-Cheng Tai, Jiang Yang. A new variational approach based on level-set function for convex hull problem with outliers. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020070

[5]

Shenglan Xie, Maoan Han, Peng Zhu. A posteriori error estimate of weak Galerkin fem for second order elliptic problem with mixed boundary condition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020340

[6]

Yifan Chen, Thomas Y. Hou. Function approximation via the subsampled Poincaré inequality. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 169-199. doi: 10.3934/dcds.2020296

[7]

Andrew D. Lewis. Erratum for "nonholonomic and constrained variational mechanics". Journal of Geometric Mechanics, 2020, 12 (4) : 671-675. doi: 10.3934/jgm.2020033

[8]

Predrag S. Stanimirović, Branislav Ivanov, Haifeng Ma, Dijana Mosić. A survey of gradient methods for solving nonlinear optimization. Electronic Research Archive, 2020, 28 (4) : 1573-1624. doi: 10.3934/era.2020115

[9]

Christopher S. Goodrich, Benjamin Lyons, Mihaela T. Velcsov. Analytical and numerical monotonicity results for discrete fractional sequential differences with negative lower bound. Communications on Pure & Applied Analysis, 2021, 20 (1) : 339-358. doi: 10.3934/cpaa.2020269

[10]

Shipra Singh, Aviv Gibali, Xiaolong Qin. Cooperation in traffic network problems via evolutionary split variational inequalities. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020170

[11]

Eduard Feireisl, Elisabetta Rocca, Giulio Schimperna, Arghir Zarnescu. Weak sequential stability for a nonlinear model of nematic electrolytes. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 219-241. doi: 10.3934/dcdss.2020366

[12]

Huiying Fan, Tao Ma. Parabolic equations involving Laguerre operators and weighted mixed-norm estimates. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5487-5508. doi: 10.3934/cpaa.2020249

[13]

Juan Pablo Pinasco, Mauro Rodriguez Cartabia, Nicolas Saintier. Evolutionary game theory in mixed strategies: From microscopic interactions to kinetic equations. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2020051

[14]

Jie Zhang, Yuping Duan, Yue Lu, Michael K. Ng, Huibin Chang. Bilinear constraint based ADMM for mixed Poisson-Gaussian noise removal. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020071

[15]

Min Chen, Olivier Goubet, Shenghao Li. Mathematical analysis of bump to bucket problem. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5567-5580. doi: 10.3934/cpaa.2020251

[16]

Qingfang Wang, Hua Yang. Solutions of nonlocal problem with critical exponent. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5591-5608. doi: 10.3934/cpaa.2020253

[17]

Reza Chaharpashlou, Abdon Atangana, Reza Saadati. On the fuzzy stability results for fractional stochastic Volterra integral equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020432

[18]

Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020450

[19]

M. S. Lee, H. G. Harno, B. S. Goh, K. H. Lim. On the bang-bang control approach via a component-wise line search strategy for unconstrained optimization. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 45-61. doi: 10.3934/naco.2020014

[20]

Mohammed Abdulrazaq Kahya, Suhaib Abduljabbar Altamir, Zakariya Yahya Algamal. Improving whale optimization algorithm for feature selection with a time-varying transfer function. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 87-98. doi: 10.3934/naco.2020017

2019 Impact Factor: 1.366

Metrics

  • PDF downloads (35)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]