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 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.

[1]

X. X. Huang, Xiaoqi Yang. Levitin-Polyak well-posedness in generalized variational inequality problems with functional constraints. Journal of Industrial and Management Optimization, 2007, 3 (4) : 671-684. doi: 10.3934/jimo.2007.3.671

[2]

Rong Hu, Ya-Ping Fang, Nan-Jing Huang. Levitin-Polyak well-posedness for variational inequalities and for optimization problems with variational inequality constraints. Journal of Industrial and Management Optimization, 2010, 6 (3) : 465-481. doi: 10.3934/jimo.2010.6.465

[3]

M. H. Li, S. J. Li, W. Y. Zhang. Levitin-Polyak well-posedness of generalized vector quasi-equilibrium problems. Journal of Industrial and Management Optimization, 2009, 5 (4) : 683-696. doi: 10.3934/jimo.2009.5.683

[4]

Abd-semii Oluwatosin-Enitan Owolabi, Timilehin Opeyemi Alakoya, Adeolu Taiwo, Oluwatosin Temitope Mewomo. A new inertial-projection algorithm for approximating common solution of variational inequality and fixed point problems of multivalued mappings. Numerical Algebra, Control and Optimization, 2022, 12 (2) : 255-278. doi: 10.3934/naco.2021004

[5]

Suxiang He, Pan Zhang, Xiao Hu, Rong Hu. A sample average approximation method based on a D-gap function for stochastic variational inequality problems. Journal of Industrial and Management Optimization, 2014, 10 (3) : 977-987. doi: 10.3934/jimo.2014.10.977

[6]

Kenji Kimura, Yeong-Cheng Liou, Soon-Yi Wu, Jen-Chih Yao. Well-posedness for parametric vector equilibrium problems with applications. Journal of Industrial and Management Optimization, 2008, 4 (2) : 313-327. doi: 10.3934/jimo.2008.4.313

[7]

Nan-Jing Huang, Xian-Jun Long, Chang-Wen Zhao. Well-Posedness for vector quasi-equilibrium problems with applications. Journal of Industrial and Management Optimization, 2009, 5 (2) : 341-349. doi: 10.3934/jimo.2009.5.341

[8]

Masao Fukushima. A class of gap functions for quasi-variational inequality problems. Journal of Industrial and Management Optimization, 2007, 3 (2) : 165-171. doi: 10.3934/jimo.2007.3.165

[9]

Jiawei Chen, Zhongping Wan, Liuyang Yuan. Existence of solutions and $\alpha$-well-posedness for a system of constrained set-valued variational inequalities. Numerical Algebra, Control and Optimization, 2013, 3 (3) : 567-581. doi: 10.3934/naco.2013.3.567

[10]

Zhichun Zhai. Well-posedness for two types of generalized Keller-Segel system of chemotaxis in critical Besov spaces. Communications on Pure and Applied Analysis, 2011, 10 (1) : 287-308. doi: 10.3934/cpaa.2011.10.287

[11]

Vanessa Barros, Felipe Linares. A remark on the well-posedness of a degenerated Zakharov system. Communications on Pure and Applied Analysis, 2015, 14 (4) : 1259-1274. doi: 10.3934/cpaa.2015.14.1259

[12]

Carlos F. Daganzo. On the variational theory of traffic flow: well-posedness, duality and applications. Networks and Heterogeneous Media, 2006, 1 (4) : 601-619. doi: 10.3934/nhm.2006.1.601

[13]

Wenyan Zhang, Shu Xu, Shengji Li, Xuexiang Huang. Generalized weak sharp minima of variational inequality problems with functional constraints. Journal of Industrial and Management Optimization, 2013, 9 (3) : 621-630. doi: 10.3934/jimo.2013.9.621

[14]

Zhaohui Huo, Boling Guo. The well-posedness of Cauchy problem for the generalized nonlinear dispersive equation. Discrete and Continuous Dynamical Systems, 2005, 12 (3) : 387-402. doi: 10.3934/dcds.2005.12.387

[15]

Yongye Zhao, Yongsheng Li, Wei Yan. Local Well-posedness and Persistence Property for the Generalized Novikov Equation. Discrete and Continuous Dynamical Systems, 2014, 34 (2) : 803-820. doi: 10.3934/dcds.2014.34.803

[16]

Lam Quoc Anh, Nguyen Van Hung. Gap functions and Hausdorff continuity of solution mappings to parametric strong vector quasiequilibrium problems. Journal of Industrial and Management Optimization, 2018, 14 (1) : 65-79. doi: 10.3934/jimo.2017037

[17]

C. R. Chen, S. J. Li. Semicontinuity of the solution set map to a set-valued weak vector variational inequality. Journal of Industrial and Management Optimization, 2007, 3 (3) : 519-528. doi: 10.3934/jimo.2007.3.519

[18]

Alexander V. Rezounenko, Petr Zagalak. Non-local PDEs with discrete state-dependent delays: Well-posedness in a metric space. Discrete and Continuous Dynamical Systems, 2013, 33 (2) : 819-835. doi: 10.3934/dcds.2013.33.819

[19]

Mohammad Eslamian, Ahmad Kamandi. A novel algorithm for approximating common solution of a system of monotone inclusion problems and common fixed point problem. Journal of Industrial and Management Optimization, 2021  doi: 10.3934/jimo.2021210

[20]

Qilin Wang, Shengji Li. Semicontinuity of approximate solution mappings to generalized vector equilibrium problems. Journal of Industrial and Management Optimization, 2016, 12 (4) : 1303-1309. doi: 10.3934/jimo.2016.12.1303

2020 Impact Factor: 1.801

Metrics

  • PDF downloads (92)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]