American Institute of Mathematical Sciences

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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

Jian-Wen Peng 1, and  Xin-Min Yang 1,

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.
Keywords: metric characterization., Levitin-Polyak approximating solution sequence, Levitin-Polyak well-posedness, System of generalized vector variational inequality problems, gap function.
Mathematics Subject Classification: Primary: 49K40, 90C31; Secondary: 47J2.
Citation: Jian-Wen Peng, Xin-Min Yang. Levitin-Polyak well-posedness of a system of generalized vector variational inequality problems. Journal of Industrial and 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), 547-557. doi: 10.1023/A:1026495115191.

[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.

[3]

J. P. Aubin and I. Ekeland, Applied Nonlinear Analisis, John Wiley & Sons, 1984.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[11]

S. Deng, Coercivity properties and well-posedness in vector optimization, RAIRO Oper. Res., 37 (2003), 195-208. doi: 10.1051/ro:2003021.

[12]

A. L. Dontchev and T. Zolezzi, Well-posed Optimization Problems, Springer-Verlag , 1993.

[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.

[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.

[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.

[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}

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[26]

C. Kuratowski, Topologie, Panstwove Wydanictwo Naukowe, Warszawa, Poland, 1952.

[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.

[28]

E. S. Levitin and B. T. Polyak, Convergence of minimizing sequences in conditional extremum problem, Soviet Mathematics Doklady, 7 (1966), 764-767.

[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.

[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.

[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.

[32]

D. T. Luc, Theory of Vector Optimization, Springer, Berlin, 1989.

[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.

[34]

R. Lucchetti, Convexity and Well-posed Problems, springer, 2006.

[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.

[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.

[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.

[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.

[39]

T. Zolezzi, Extended well-posedness of optimization problems, J. Optim. Theory Appl., 91 (1996), 257-266. doi: 10.1007/BF02192292.

show all references

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.

[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.

[3]

J. P. Aubin and I. Ekeland, Applied Nonlinear Analisis, John Wiley & Sons, 1984.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[11]

S. Deng, Coercivity properties and well-posedness in vector optimization, RAIRO Oper. Res., 37 (2003), 195-208. doi: 10.1051/ro:2003021.

[12]

A. L. Dontchev and T. Zolezzi, Well-posed Optimization Problems, Springer-Verlag , 1993.

[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.

[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.

[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.

[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}

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[26]

C. Kuratowski, Topologie, Panstwove Wydanictwo Naukowe, Warszawa, Poland, 1952.

[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.

[28]

E. S. Levitin and B. T. Polyak, Convergence of minimizing sequences in conditional extremum problem, Soviet Mathematics Doklady, 7 (1966), 764-767.

[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.

[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.

[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.

[32]

D. T. Luc, Theory of Vector Optimization, Springer, Berlin, 1989.

[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.

[34]

R. Lucchetti, Convexity and Well-posed Problems, springer, 2006.

[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.

[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.

[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.

[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.

[39]

T. Zolezzi, Extended well-posedness of optimization problems, J. Optim. Theory Appl., 91 (1996), 257-266. doi: 10.1007/BF02192292.

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