American Institute of Mathematical Sciences

Advanced
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 & 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.  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

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.  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

[1]

X. X. Huang, Xiaoqi Yang. Levitin-Polyak well-posedness in generalized variational inequality problems with functional constraints. Journal of Industrial & 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 & 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 & 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 & Optimization, 2021  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 & 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 & 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 & 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 & 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 & 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 & 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 & 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 & Heterogeneous Media, 2006, 1 (4) : 601-619. doi: 10.3934/nhm.2006.1.601
[13]

Zhaohui Huo, Boling Guo. The well-posedness of Cauchy problem for the generalized nonlinear dispersive equation. Discrete & Continuous Dynamical Systems, 2005, 12 (3) : 387-402. doi: 10.3934/dcds.2005.12.387
[14]

Yongye Zhao, Yongsheng Li, Wei Yan. Local Well-posedness and Persistence Property for the Generalized Novikov Equation. Discrete & Continuous Dynamical Systems, 2014, 34 (2) : 803-820. doi: 10.3934/dcds.2014.34.803
[15]

Wenyan Zhang, Shu Xu, Shengji Li, Xuexiang Huang. Generalized weak sharp minima of variational inequality problems with functional constraints. Journal of Industrial & Management Optimization, 2013, 9 (3) : 621-630. doi: 10.3934/jimo.2013.9.621
[16]

Alexander V. Rezounenko, Petr Zagalak. Non-local PDEs with discrete state-dependent delays: Well-posedness in a metric space. Discrete & Continuous Dynamical Systems, 2013, 33 (2) : 819-835. doi: 10.3934/dcds.2013.33.819
[17]

Lam Quoc Anh, Nguyen Van Hung. Gap functions and Hausdorff continuity of solution mappings to parametric strong vector quasiequilibrium problems. Journal of Industrial & Management Optimization, 2018, 14 (1) : 65-79. doi: 10.3934/jimo.2017037
[18]

C. R. Chen, S. J. Li. Semicontinuity of the solution set map to a set-valued weak vector variational inequality. Journal of Industrial & Management Optimization, 2007, 3 (3) : 519-528. doi: 10.3934/jimo.2007.3.519
[19]

Iñigo U. Erneta. Well-posedness for boundary value problems for coagulation-fragmentation equations. Kinetic & Related Models, 2020, 13 (4) : 815-835. doi: 10.3934/krm.2020028
[20]

Lam Quoc Anh, Pham Thanh Duoc, Tran Quoc Duy. Existence and well-posedness for excess demand equilibrium problems. Numerical Algebra, Control & Optimization, 2021  doi: 10.3934/naco.2021043

2020 Impact Factor: 1.801

Tools

Metrics

  • PDF downloads (69)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences