# American Institute of Mathematical Sciences

July  2015, 11(3): 701-714. doi: 10.3934/jimo.2015.11.701

## Levitin-Polyak well-posedness of a system of generalized vector variational inequality problems

 1 School of Mathematics, Chongqing Normal University, Chongqing 400047, China, China

Received  February 2012 Revised  May 2014 Published  October 2014

In this paper, we introduce two types of Levitin-Polyak well-posedness for a system of generalized vector variational inequality problems. By means of a gap function of the system of generalized vector variational inequality problems, we establish equivalence between the two types of Levitin-Polyak well-posedness of the system of generalized vector variational inequality problems and the corresponding well-posednesses of the minimization problems. We also present some metric characterizations for the two types of Levitin-Polyak well-posedness of the system of generalized vector variational inequality problems. The results in this paper generalize, extend and improve some known results in the literature.
Citation: Jian-Wen Peng, Xin-Min Yang. Levitin-Polyak well-posedness of a system of generalized vector variational inequality problems. Journal of Industrial & Management Optimization, 2015, 11 (3) : 701-714. doi: 10.3934/jimo.2015.11.701
##### References:
 [1] Q. H. Ansari, S. Schaible and J. C. Yao, Systems of Vector Equilibrium problems and its applications,, J. Optim. Theory and Appl., 107 (2000), 547.  doi: 10.1023/A:1026495115191.  Google Scholar [2] Q. H. Ansari and J. C. Yao, A fixed-point theorem and its applications to the Systems of variational inequalities,, Bull. Austr. Math. Soc., 59 (1999), 433.  doi: 10.1017/S0004972700033116.  Google Scholar [3] J. P. Aubin and I. Ekeland, Applied Nonlinear Analisis,, John Wiley & Sons, (1984).   Google Scholar [4] E. Bednarczuk, Well-posedness of vector optimization problems,, in Lecture Notes in Economics and Mathematical Systems, 294 (1987), 51.  doi: 10.1007/978-3-642-46618-2_2.  Google Scholar [5] M. Bianchi, Pseudo P-monotone Operators and Variational Inequalities,, Report 6, (1993).   Google Scholar [6] L. C. Ceng and J. C. Yao, Well-posedness of generalized mixed variational inequalities, inclusion problems and fixed-point problems,, Nonlinear Analysis, 69 (2008), 4585.  doi: 10.1016/j.na.2007.11.015.  Google Scholar [7] G. Y. Chen, X. X. Huang and X. Q. Yang, Vector Optimization, Set-valued and Variational Analysis,, Lecture notes in economics and mathematical systems. Springer, (2005).   Google Scholar [8] G. Cohen and F. Chaplais, Nested monotony for variational inequalities over a product of spaces and convergence of iterative algorithms,, J. Optim. Theory and Appl., 59 (1988), 369.  doi: 10.1007/BF00940305.  Google Scholar [9] G. P. Crespi, A. Guerraggio and M. Rocca, Well Posedness in Vector Optimization Problems and Vector Variational Inequalities,, J. Optim. Theory and Appl., 132 (2007), 213.  doi: 10.1007/s10957-006-9144-2.  Google Scholar [10] G. P. Crespi, M. Papalia and M. Rocca, Extended Well-Posedness of Quasiconvex Vector Optimization Problems,, J. Optim. Theory and Appl., 141 (2009), 285.  doi: 10.1007/s10957-008-9494-z.  Google Scholar [11] S. Deng, Coercivity properties and well-posedness in vector optimization,, RAIRO Oper. Res., 37 (2003), 195.  doi: 10.1051/ro:2003021.  Google Scholar [12] A. L. Dontchev and T. Zolezzi, Well-posed Optimization Problems,, Springer-Verlag, (1993).   Google Scholar [13] F. Giannessi, Theorems alternative, Quadratic programs, and complementarity problems,, In variational inequalities and complementarity problems, (1980), 151.   Google Scholar [14] Y. P. Fang and R. Hu, Parametric well-posedness for variational inequalities defined by bifunctions,, Computers and Mathematics with Applications, 53 (2007), 1306.  doi: 10.1016/j.camwa.2006.09.009.  Google Scholar [15] Y. P. Fang, N. J. Huang and J. C. Yao, Well-posedness of mixed variational inequalities, inclusion problems and fixed point problems,, J. Glob. Optim., 41 (2008), 117.  doi: 10.1007/s10898-007-9169-6.  Google Scholar [16] M. Furi and A. Vignoli, About well-posed optimization problems for functions in metric spaces,, J. Optim. Theory Appl., 5 (1970), 223.   Google Scholar [17] R. Hu and Y. P. Fang, Levitin-Polyak well-posedness of variational inequalities,, Nonlinear Analysis, 72 (2010), 373.  doi: 10.1016/j.na.2009.06.071.  Google Scholar [18] X. X. Huang, Extended well-posed properties of vector optimization problems,, J. Optim. Theory and Appl., 106 (2000), 165.  doi: 10.1023/A:1004615325743.  Google Scholar [19] X. X. Huang, Extended and strongly extended well-posed properties of set-valued optimization problems,, Math. Meth. Oper. Res., 53 (2001), 101.  doi: 10.1007/s001860000100.  Google Scholar [20] X. X. Huang and X. Q. Yang, Generalized Levitin-Polyak well-posedness in constrained optimization,, SIAM J. Optim., 17 (2006), 243.  doi: 10.1137/040614943.  Google Scholar [21] X. X. Huang and X. Q. Yang, Levitin-Polyak well-posedness of constrained vector optimization problems,, J. Glob. Optim., 37 (2007), 287.  doi: 10.1007/s10898-006-9050-z.  Google Scholar [22] X. X. Huang and X. Q. Yang, Levitin-Polyak well-posedness of vector variational inequality problems with functional constraints,, Numer. Funct. Anal. Optim., 31 (2010), 440.  doi: 10.1080/01630563.2010.485296.  Google Scholar [23] X. X. Huang and X. Q. Yang, Levitin-Polyak well-posedness in generalized variational inequalities problems with functional constraints,, J. Ind. Manag. Optim., 3 (2007), 671.  doi: 10.3934/jimo.2007.3.671.  Google Scholar [24] X. X. Huang, X. Q. Yang and D. L. Zhu, Levitin-Polyak well-posedness of variational inequalities problems with functional constraints,, J. Glob. Optim., 44 (2009), 159.  doi: 10.1007/s10898-008-9310-1.  Google Scholar [25] A. S. Konsulova and J. P. Revalski, Constrained convex optomization problems-well-posedness and stability,, Numer. Funct. Anal. Optim., 15 (1994), 889.  doi: 10.1080/01630569408816598.  Google Scholar [26] C. Kuratowski, Topologie, Panstwove Wydanictwo Naukowe,, Warszawa, (1952).   Google Scholar [27] C. S. Lalitha and G. Bhatia, well-posedness for variational inequality problems with generalized monotone set-valued maps,, Numer. Funct. Anal. Optim., 30 (2009), 548.  doi: 10.1080/01630560902987972.  Google Scholar [28] E. S. Levitin and B. T. Polyak, Convergence of minimizing sequences in conditional extremum problem,, Soviet Mathematics Doklady, 7 (1966), 764.   Google Scholar [29] M. H. Li, S. J. Li and W. Y. Zhang, Levitin-Polyak well-posedness of generalized vector quasi-equilibrium problems,, J. Ind. Manag. Optim., 5 (2009), 683.  doi: 10.3934/jimo.2009.5.683.  Google Scholar [30] M. B. Lignola and J. Morgan, Approximating solutions and $\alpha$-well-posedness for variational inequalities and Nash equilibria,, in Decision and Control in Management Science, 4 (2002), 367.  doi: 10.1007/978-1-4757-3561-1_20.  Google Scholar [31] P. Loridan, Well-posed vector optimization, recent developments in well-posed variational problems,, Mathematics and its Applications, 331 (1995), 171.   Google Scholar [32] D. T. Luc, Theory of Vector Optimization,, Springer, (1989).   Google Scholar [33] R. Lucchetti, Well-posedness towards vector optimization},, Lecture Notes in Economics and Mathematical Systems, 294 (1987), 194.  doi: 10.1007/978-3-642-46618-2_13.  Google Scholar [34] R. Lucchetti, Convexity and Well-posed Problems,, springer, (2006).   Google Scholar [35] R. Lucchetti and F. Patrone, A characterization of Tykhonov well-posedness for minimum problems with applications to variational inequalities,, Numer. Funct. Anal. Optim., 3 (1981), 461.  doi: 10.1080/01630568108816100.  Google Scholar [36] J. S. Pang, Asymmetric variational inequality problems over product sets: Applications and iterative methods,, Mathematical Programming, 31 (1985), 206.  doi: 10.1007/BF02591749.  Google Scholar [37] A. N. Tykhonov, On the stability of the functional optimization problem,, USSRJ. Comput. Math. Math. Phys., 6 (1966), 28.  doi: 10.1016/0041-5553(66)90003-6.  Google Scholar [38] Z. Xu, D. L. Zhu and X. X. Huang, Levitin-Polyak well-posedness in generalized vector variational inequality problem with functional constraints,, Math. Meth. Oper. Res., 67 (2008), 505.  doi: 10.1007/s00186-007-0200-y.  Google Scholar [39] T. Zolezzi, Extended well-posedness of optimization problems,, J. Optim. Theory Appl., 91 (1996), 257.  doi: 10.1007/BF02192292.  Google Scholar

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##### References:
 [1] Q. H. Ansari, S. Schaible and J. C. Yao, Systems of Vector Equilibrium problems and its applications,, J. Optim. Theory and Appl., 107 (2000), 547.  doi: 10.1023/A:1026495115191.  Google Scholar [2] Q. H. Ansari and J. C. Yao, A fixed-point theorem and its applications to the Systems of variational inequalities,, Bull. Austr. Math. Soc., 59 (1999), 433.  doi: 10.1017/S0004972700033116.  Google Scholar [3] J. P. Aubin and I. Ekeland, Applied Nonlinear Analisis,, John Wiley & Sons, (1984).   Google Scholar [4] E. Bednarczuk, Well-posedness of vector optimization problems,, in Lecture Notes in Economics and Mathematical Systems, 294 (1987), 51.  doi: 10.1007/978-3-642-46618-2_2.  Google Scholar [5] M. Bianchi, Pseudo P-monotone Operators and Variational Inequalities,, Report 6, (1993).   Google Scholar [6] L. C. Ceng and J. C. Yao, Well-posedness of generalized mixed variational inequalities, inclusion problems and fixed-point problems,, Nonlinear Analysis, 69 (2008), 4585.  doi: 10.1016/j.na.2007.11.015.  Google Scholar [7] G. Y. Chen, X. X. Huang and X. Q. Yang, Vector Optimization, Set-valued and Variational Analysis,, Lecture notes in economics and mathematical systems. Springer, (2005).   Google Scholar [8] G. Cohen and F. Chaplais, Nested monotony for variational inequalities over a product of spaces and convergence of iterative algorithms,, J. Optim. Theory and Appl., 59 (1988), 369.  doi: 10.1007/BF00940305.  Google Scholar [9] G. P. Crespi, A. Guerraggio and M. Rocca, Well Posedness in Vector Optimization Problems and Vector Variational Inequalities,, J. Optim. Theory and Appl., 132 (2007), 213.  doi: 10.1007/s10957-006-9144-2.  Google Scholar [10] G. P. Crespi, M. Papalia and M. Rocca, Extended Well-Posedness of Quasiconvex Vector Optimization Problems,, J. Optim. Theory and Appl., 141 (2009), 285.  doi: 10.1007/s10957-008-9494-z.  Google Scholar [11] S. Deng, Coercivity properties and well-posedness in vector optimization,, RAIRO Oper. Res., 37 (2003), 195.  doi: 10.1051/ro:2003021.  Google Scholar [12] A. L. Dontchev and T. Zolezzi, Well-posed Optimization Problems,, Springer-Verlag, (1993).   Google Scholar [13] F. Giannessi, Theorems alternative, Quadratic programs, and complementarity problems,, In variational inequalities and complementarity problems, (1980), 151.   Google Scholar [14] Y. P. Fang and R. Hu, Parametric well-posedness for variational inequalities defined by bifunctions,, Computers and Mathematics with Applications, 53 (2007), 1306.  doi: 10.1016/j.camwa.2006.09.009.  Google Scholar [15] Y. P. Fang, N. J. Huang and J. C. Yao, Well-posedness of mixed variational inequalities, inclusion problems and fixed point problems,, J. Glob. Optim., 41 (2008), 117.  doi: 10.1007/s10898-007-9169-6.  Google Scholar [16] M. Furi and A. Vignoli, About well-posed optimization problems for functions in metric spaces,, J. Optim. Theory Appl., 5 (1970), 223.   Google Scholar [17] R. Hu and Y. P. Fang, Levitin-Polyak well-posedness of variational inequalities,, Nonlinear Analysis, 72 (2010), 373.  doi: 10.1016/j.na.2009.06.071.  Google Scholar [18] X. X. Huang, Extended well-posed properties of vector optimization problems,, J. Optim. Theory and Appl., 106 (2000), 165.  doi: 10.1023/A:1004615325743.  Google Scholar [19] X. X. Huang, Extended and strongly extended well-posed properties of set-valued optimization problems,, Math. Meth. Oper. Res., 53 (2001), 101.  doi: 10.1007/s001860000100.  Google Scholar [20] X. X. Huang and X. Q. Yang, Generalized Levitin-Polyak well-posedness in constrained optimization,, SIAM J. Optim., 17 (2006), 243.  doi: 10.1137/040614943.  Google Scholar [21] X. X. Huang and X. Q. Yang, Levitin-Polyak well-posedness of constrained vector optimization problems,, J. Glob. Optim., 37 (2007), 287.  doi: 10.1007/s10898-006-9050-z.  Google Scholar [22] X. X. Huang and X. Q. Yang, Levitin-Polyak well-posedness of vector variational inequality problems with functional constraints,, Numer. Funct. Anal. Optim., 31 (2010), 440.  doi: 10.1080/01630563.2010.485296.  Google Scholar [23] X. X. Huang and X. Q. Yang, Levitin-Polyak well-posedness in generalized variational inequalities problems with functional constraints,, J. Ind. Manag. Optim., 3 (2007), 671.  doi: 10.3934/jimo.2007.3.671.  Google Scholar [24] X. X. Huang, X. Q. Yang and D. L. Zhu, Levitin-Polyak well-posedness of variational inequalities problems with functional constraints,, J. Glob. Optim., 44 (2009), 159.  doi: 10.1007/s10898-008-9310-1.  Google Scholar [25] A. S. Konsulova and J. P. Revalski, Constrained convex optomization problems-well-posedness and stability,, Numer. Funct. Anal. Optim., 15 (1994), 889.  doi: 10.1080/01630569408816598.  Google Scholar [26] C. Kuratowski, Topologie, Panstwove Wydanictwo Naukowe,, Warszawa, (1952).   Google Scholar [27] C. S. Lalitha and G. Bhatia, well-posedness for variational inequality problems with generalized monotone set-valued maps,, Numer. Funct. Anal. Optim., 30 (2009), 548.  doi: 10.1080/01630560902987972.  Google Scholar [28] E. S. Levitin and B. T. Polyak, Convergence of minimizing sequences in conditional extremum problem,, Soviet Mathematics Doklady, 7 (1966), 764.   Google Scholar [29] M. H. Li, S. J. Li and W. Y. Zhang, Levitin-Polyak well-posedness of generalized vector quasi-equilibrium problems,, J. Ind. Manag. Optim., 5 (2009), 683.  doi: 10.3934/jimo.2009.5.683.  Google Scholar [30] M. B. Lignola and J. Morgan, Approximating solutions and $\alpha$-well-posedness for variational inequalities and Nash equilibria,, in Decision and Control in Management Science, 4 (2002), 367.  doi: 10.1007/978-1-4757-3561-1_20.  Google Scholar [31] P. Loridan, Well-posed vector optimization, recent developments in well-posed variational problems,, Mathematics and its Applications, 331 (1995), 171.   Google Scholar [32] D. T. Luc, Theory of Vector Optimization,, Springer, (1989).   Google Scholar [33] R. Lucchetti, Well-posedness towards vector optimization},, Lecture Notes in Economics and Mathematical Systems, 294 (1987), 194.  doi: 10.1007/978-3-642-46618-2_13.  Google Scholar [34] R. Lucchetti, Convexity and Well-posed Problems,, springer, (2006).   Google Scholar [35] R. Lucchetti and F. Patrone, A characterization of Tykhonov well-posedness for minimum problems with applications to variational inequalities,, Numer. Funct. Anal. Optim., 3 (1981), 461.  doi: 10.1080/01630568108816100.  Google Scholar [36] J. S. Pang, Asymmetric variational inequality problems over product sets: Applications and iterative methods,, Mathematical Programming, 31 (1985), 206.  doi: 10.1007/BF02591749.  Google Scholar [37] A. N. Tykhonov, On the stability of the functional optimization problem,, USSRJ. Comput. Math. Math. Phys., 6 (1966), 28.  doi: 10.1016/0041-5553(66)90003-6.  Google Scholar [38] Z. Xu, D. L. Zhu and X. X. Huang, Levitin-Polyak well-posedness in generalized vector variational inequality problem with functional constraints,, Math. Meth. Oper. Res., 67 (2008), 505.  doi: 10.1007/s00186-007-0200-y.  Google Scholar [39] T. Zolezzi, Extended well-posedness of optimization problems,, J. Optim. Theory Appl., 91 (1996), 257.  doi: 10.1007/BF02192292.  Google Scholar
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