# 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:
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Yao, Well-posedness of generalized mixed variational inequalities, inclusion problems and fixed-point problems, Nonlinear Analysis, TMA, 69 (2008), 4585-4603. 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, Berlin 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), 369-390. 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), 213-226. 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), 285-297. 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), 195-208. 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, (Edited by R. W. Cottle, F. Giannessi, and J. L. Lions), John Wiley and Sins, New york, (1980), 151-186.  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), 1306-1316. 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), 117-133. 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), 223-229. \vspace*{2pt}  Google Scholar [17] R. Hu and Y. P. Fang, Levitin-Polyak well-posedness of variational inequalities, Nonlinear Analysis, TMA, 72 (2010), 373-381. 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), 165-182. 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), 101-116. 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), 243-258. 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), 287-304. 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), 440-459. 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), 671-684. 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), 159-174. 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), 889-907. doi: 10.1080/01630569408816598.  Google Scholar [26] C. Kuratowski, Topologie, Panstwove Wydanictwo Naukowe, Warszawa, Poland, 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), 548-565. 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), 764-767. 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), 683-696. 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, Kluwer Academic Publishers, 4 (2002), 367-377. 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, Kluwer Academic Publishers, Dordrecht, 331 (1995), 171-192.  Google Scholar [32] D. T. Luc, Theory of Vector Optimization, Springer, Berlin, 1989.  Google Scholar [33] R. Lucchetti, Well-posedness towards vector optimization}, Lecture Notes in Economics and Mathematical Systems, Springer-Verlag, Berlin, 294 (1987), 194-207. 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), 461-476. 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), 206-219. 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), 28-33. 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), 505-524. doi: 10.1007/s00186-007-0200-y.  Google Scholar [39] T. Zolezzi, Extended well-posedness of optimization problems, J. Optim. Theory Appl., 91 (1996), 257-266. 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), 547-557. 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), 433-442. 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, Springer-Verlag, Berlin, 294 (1987), 51-61. doi: 10.1007/978-3-642-46618-2_2.  Google Scholar [5] M. Bianchi, Pseudo P-monotone Operators and Variational Inequalities, Report 6, Istituto di econometria e Matematica per le decisioni economiche, Universita Cattolica del Sacro Cuore, Milan, Italy, 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, TMA, 69 (2008), 4585-4603. 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, Berlin 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), 369-390. 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), 213-226. 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), 285-297. 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), 195-208. 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, (Edited by R. W. Cottle, F. Giannessi, and J. L. Lions), John Wiley and Sins, New york, (1980), 151-186.  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), 1306-1316. 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), 117-133. 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), 223-229. \vspace*{2pt}  Google Scholar [17] R. Hu and Y. P. Fang, Levitin-Polyak well-posedness of variational inequalities, Nonlinear Analysis, TMA, 72 (2010), 373-381. 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), 165-182. 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), 101-116. 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), 243-258. 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), 287-304. 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), 440-459. 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), 671-684. 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), 159-174. 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), 889-907. doi: 10.1080/01630569408816598.  Google Scholar [26] C. Kuratowski, Topologie, Panstwove Wydanictwo Naukowe, Warszawa, Poland, 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), 548-565. 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), 764-767. 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), 683-696. 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, Kluwer Academic Publishers, 4 (2002), 367-377. 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, Kluwer Academic Publishers, Dordrecht, 331 (1995), 171-192.  Google Scholar [32] D. T. Luc, Theory of Vector Optimization, Springer, Berlin, 1989.  Google Scholar [33] R. Lucchetti, Well-posedness towards vector optimization}, Lecture Notes in Economics and Mathematical Systems, Springer-Verlag, Berlin, 294 (1987), 194-207. 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), 461-476. 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), 206-219. 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), 28-33. 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), 505-524. doi: 10.1007/s00186-007-0200-y.  Google Scholar [39] T. Zolezzi, Extended well-posedness of optimization problems, J. Optim. Theory Appl., 91 (1996), 257-266. doi: 10.1007/BF02192292.  Google Scholar
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