• Previous Article
    An unconstrained optimization approach for finding real eigenvalues of even order symmetric tensors
  • NACO Home
  • This Issue
  • Next Article
    A convergence theorem of common fixed points of a countably infinite family of asymptotically quasi-$f_i$-expansive mappings in convex metric spaces
2013, 3(3): 567-581. doi: 10.3934/naco.2013.3.567

Existence of solutions and $\alpha$-well-posedness for a system of constrained set-valued variational inequalities

1. 

School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China

2. 

School of Mathematics and Statistics, Wuhan University, Wuhan, 430072

Received  September 2011 Revised  April 2013 Published  July 2013

The notions of $\alpha$-well-posedness and generalized $\alpha$-well-posedness for a system of constrained variational inequalities involving set-valued mappings (for short, (SCVI)) are introduced in Hilbert spaces. Existence theorems of solutions for (SCVI) are established by using penalty techniques. Metric characterizations of $\alpha$-well-posedness and generalized $\alpha$-well-posedness, in terms of the approximate solutions sets, are presented. Finally, the equivalences between (generalized) $\alpha$-well-posedness for (SCVI) and existence and uniqueness of its solutions are also derived under quite mild assumptions.
Citation: 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
References:
[1]

R. T. Agarwal, N. J. Huang and Y. J. Cho, Generalized nonlinear mixed implicit quasi-variational inclusions with set-valued mappings, J. Inequal. Appl., 7 (2002), 807-828.

[2]

H. Attouch, "E.D.P.associées à de sous-différentiels," Thèse de Doctorat d'état ES Sciences Mathématiques, Université Paris 6, 1976.

[3]

L. C. Ceng, N. Hadjisavvas, S. Schaible and J. C. Yao, Well-posedness for mixed quasivariational-like inequalities, J. Optim. Theory Appl., 139 (2008), 109-125. doi: 10.1007/s10957-008-9428-9.

[4]

J. W. Chen, Z. Wan and Y. J. Cho, Levitin-Polyak well-posedness by perturbations for systems of set-valued vector quasi-equilibrium problems, Math. Meth. Oper. Res., 77 (2013), 33-64. doi: 10.1007/s00186-012-0414-5.

[5]

J. W. Chen and Z. Wan, Existence of solutions and convergence analysis for a system of quasivariational inclusions in Banach spaces, J. Inequal. Appl., 49 (2011). doi: 10.1186/1029-242X-2011-49.

[6]

Y. J. Cho, Y. P. Fang, N. J. Huang and N. J. Hwang, Algorithms for systems of nonlinear variational inequalities, J. Korean Math. Soc., 41 (2004), 203-210.

[7]

Y. P. Fang, R. Hu and N. J. Huang, Well-posedness for equilibrium problems and for optimization problems with equilibrium constraints, Comput. Math. Appl., 55 (2008), 89-100. doi: 10.1016/j.camwa.2007.03.019.

[8]

M. Furi and A. Vignoli, About well-posed optimization problems for functions in metric spaces, J. Optim. Theory Appl., 5 (1970), 225-229. doi: 10.1007/BF00927717.

[9]

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.

[10]

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), 671-684. doi: 10.1080/01630563.2010.485296.

[11]

R. Hu, Y. P. Fang, N. J. Huang and M. M. Wong, Well-posedness of systems of equilibrium problems, Taiwanese J. Math., 14 (2010), 2435-2446.

[12]

R. Hu, Y. P. Fang and N. J. Huang, Levitin-Polyak well-posedness for variational inequalities and for optimization problems with variational inequalities, J. Ind. Manag. Optim., 6 (2010), 465-481. doi: 10.3934/jimo.2010.6.465.

[13]

G. Kassay, J. Kolumban and Z. Pales, On Nash stationary points, Publ. Math. Debrecen, 54 (1999), 267-279.

[14]

J. K. Kim and D. S. Kim, A new system of generalized nonlinear mixed variational inequalities in Hilbert spaces, J. Convex Anal., 11 (2004), 235-243.

[15]

K. Kuratowski, "Topology," (Vols. 1 and 2), Academic Press, New York, 1968.

[16]

C. S. Lalitha and G. Bhatia, Well-posedness for parametric quasivariational inequality problems and for optimizations problems with quasivariational inequality constraints, Optim., 59 (2010), 997-1011. doi: 10.1080/02331930902878358.

[17]

M. B. Lignola and J. Morgan, Well-posedness for optimization problems with constraints defined by variational inequalities having a unique solution, J. Glob. Optim., 16 (2000), 57-67. doi: 10.1023/A:1008370910807.

[18]

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, (2001), 367-378.

[19]

M. B. Lignola and J. Morgan, α-well-posedness for Nash equilibria and for optimization problems with Nash equilibrium constraints, J. Glob. Optim., 36 (2006), 439-459. doi: 10.1007/s10898-006-9020-5.

[20]

P. L. Lions, Two remarks on the convergence of convex functions and monotone operator, Nonlinear Anal., 2 (1978), 553-562.

[21]

R. Lucchetti and F. Patrone, A characterization of Tykhonov well-posedness for minimimum problems with applications to variational inequalities, Numer. Funct. Anal. Optim., 3 (1981), 461-476.

[22]

P. E. Mainge, New approach to solving a system of variational inequalities and hierarchical problems, J. Optim. Theory Appl., 138 (2008), 459-477. doi: 10.1007/s10957-008-9433-z.

[23]

A. Moudafi and M. A. Noor, Penalty method for a system of constrained variational inequalities, Optim. Lett., 6 (2012), 451-458. doi: 10.1007/s11590-010-0271-1.

[24]

M. A. Noor and K. I. Noor, Projection algorithms for solving a system of general variational inequalities, Nonlinear Anal., 70 (2009), 2700-2706. doi: 10.1016/j.na.2008.03.057.

[25]

D. Pascali and S. Sburlan, "Nonlinear Mappings of Monotone Type," Martinus Nijhoff, The Hague, 1978.

[26]

J. W. Peng and S. Y. Wu, The generalized Tykhonov well-posedness for system of vector quasi-equilibrium problems, Optim. Lett., 4 (2010), 501-512. doi: 10.1007/s11590-010-0179-9.

[27]

J. W. Peng and J. Tang, α-well-posedness for mixed quasi-variational-like inequality problems, Abstr. Appl. Anal., 2011 (2011), 1-17.

[28]

G. Stampacchia, Forms bilineaires coercivities sur les ensembles convexes, CR Acad. Sci. Paris, 258 (1964), 4413-4416.

[29]

Y. Tang and L. W. Liu, The penalty method for a new system of generalized variational inequalities, Int. J. Math. Math. Sci., 2010 (2010), 1-8. doi: 10.1155/2010/614276.

[30]

A. N. Tykhonov, On the stability of the functional optimization problem, USSR J. Comput. Math. Math. Phys., 6 (1966), 631-634.

[31]

Y. X. Yuan, Recent advances in numerical methods for nonlinear equations and nonlinear least squares, Numer. Algebra Control Optim., 1 (2011), 15-34. doi: 10.3934/naco.2011.1.15.

[32]

R. Y. Zhong and N. J. Huang, Strict feasibility for generalized mixed variational inequality in reflexive Banach spaces, Numer. Algebra Control Optim., 1 (2011), 261-274. doi: 10.3934/naco.2011.1.261.

show all references

References:
[1]

R. T. Agarwal, N. J. Huang and Y. J. Cho, Generalized nonlinear mixed implicit quasi-variational inclusions with set-valued mappings, J. Inequal. Appl., 7 (2002), 807-828.

[2]

H. Attouch, "E.D.P.associées à de sous-différentiels," Thèse de Doctorat d'état ES Sciences Mathématiques, Université Paris 6, 1976.

[3]

L. C. Ceng, N. Hadjisavvas, S. Schaible and J. C. Yao, Well-posedness for mixed quasivariational-like inequalities, J. Optim. Theory Appl., 139 (2008), 109-125. doi: 10.1007/s10957-008-9428-9.

[4]

J. W. Chen, Z. Wan and Y. J. Cho, Levitin-Polyak well-posedness by perturbations for systems of set-valued vector quasi-equilibrium problems, Math. Meth. Oper. Res., 77 (2013), 33-64. doi: 10.1007/s00186-012-0414-5.

[5]

J. W. Chen and Z. Wan, Existence of solutions and convergence analysis for a system of quasivariational inclusions in Banach spaces, J. Inequal. Appl., 49 (2011). doi: 10.1186/1029-242X-2011-49.

[6]

Y. J. Cho, Y. P. Fang, N. J. Huang and N. J. Hwang, Algorithms for systems of nonlinear variational inequalities, J. Korean Math. Soc., 41 (2004), 203-210.

[7]

Y. P. Fang, R. Hu and N. J. Huang, Well-posedness for equilibrium problems and for optimization problems with equilibrium constraints, Comput. Math. Appl., 55 (2008), 89-100. doi: 10.1016/j.camwa.2007.03.019.

[8]

M. Furi and A. Vignoli, About well-posed optimization problems for functions in metric spaces, J. Optim. Theory Appl., 5 (1970), 225-229. doi: 10.1007/BF00927717.

[9]

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.

[10]

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), 671-684. doi: 10.1080/01630563.2010.485296.

[11]

R. Hu, Y. P. Fang, N. J. Huang and M. M. Wong, Well-posedness of systems of equilibrium problems, Taiwanese J. Math., 14 (2010), 2435-2446.

[12]

R. Hu, Y. P. Fang and N. J. Huang, Levitin-Polyak well-posedness for variational inequalities and for optimization problems with variational inequalities, J. Ind. Manag. Optim., 6 (2010), 465-481. doi: 10.3934/jimo.2010.6.465.

[13]

G. Kassay, J. Kolumban and Z. Pales, On Nash stationary points, Publ. Math. Debrecen, 54 (1999), 267-279.

[14]

J. K. Kim and D. S. Kim, A new system of generalized nonlinear mixed variational inequalities in Hilbert spaces, J. Convex Anal., 11 (2004), 235-243.

[15]

K. Kuratowski, "Topology," (Vols. 1 and 2), Academic Press, New York, 1968.

[16]

C. S. Lalitha and G. Bhatia, Well-posedness for parametric quasivariational inequality problems and for optimizations problems with quasivariational inequality constraints, Optim., 59 (2010), 997-1011. doi: 10.1080/02331930902878358.

[17]

M. B. Lignola and J. Morgan, Well-posedness for optimization problems with constraints defined by variational inequalities having a unique solution, J. Glob. Optim., 16 (2000), 57-67. doi: 10.1023/A:1008370910807.

[18]

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, (2001), 367-378.

[19]

M. B. Lignola and J. Morgan, α-well-posedness for Nash equilibria and for optimization problems with Nash equilibrium constraints, J. Glob. Optim., 36 (2006), 439-459. doi: 10.1007/s10898-006-9020-5.

[20]

P. L. Lions, Two remarks on the convergence of convex functions and monotone operator, Nonlinear Anal., 2 (1978), 553-562.

[21]

R. Lucchetti and F. Patrone, A characterization of Tykhonov well-posedness for minimimum problems with applications to variational inequalities, Numer. Funct. Anal. Optim., 3 (1981), 461-476.

[22]

P. E. Mainge, New approach to solving a system of variational inequalities and hierarchical problems, J. Optim. Theory Appl., 138 (2008), 459-477. doi: 10.1007/s10957-008-9433-z.

[23]

A. Moudafi and M. A. Noor, Penalty method for a system of constrained variational inequalities, Optim. Lett., 6 (2012), 451-458. doi: 10.1007/s11590-010-0271-1.

[24]

M. A. Noor and K. I. Noor, Projection algorithms for solving a system of general variational inequalities, Nonlinear Anal., 70 (2009), 2700-2706. doi: 10.1016/j.na.2008.03.057.

[25]

D. Pascali and S. Sburlan, "Nonlinear Mappings of Monotone Type," Martinus Nijhoff, The Hague, 1978.

[26]

J. W. Peng and S. Y. Wu, The generalized Tykhonov well-posedness for system of vector quasi-equilibrium problems, Optim. Lett., 4 (2010), 501-512. doi: 10.1007/s11590-010-0179-9.

[27]

J. W. Peng and J. Tang, α-well-posedness for mixed quasi-variational-like inequality problems, Abstr. Appl. Anal., 2011 (2011), 1-17.

[28]

G. Stampacchia, Forms bilineaires coercivities sur les ensembles convexes, CR Acad. Sci. Paris, 258 (1964), 4413-4416.

[29]

Y. Tang and L. W. Liu, The penalty method for a new system of generalized variational inequalities, Int. J. Math. Math. Sci., 2010 (2010), 1-8. doi: 10.1155/2010/614276.

[30]

A. N. Tykhonov, On the stability of the functional optimization problem, USSR J. Comput. Math. Math. Phys., 6 (1966), 631-634.

[31]

Y. X. Yuan, Recent advances in numerical methods for nonlinear equations and nonlinear least squares, Numer. Algebra Control Optim., 1 (2011), 15-34. doi: 10.3934/naco.2011.1.15.

[32]

R. Y. Zhong and N. J. Huang, Strict feasibility for generalized mixed variational inequality in reflexive Banach spaces, Numer. Algebra Control Optim., 1 (2011), 261-274. doi: 10.3934/naco.2011.1.261.

[1]

Chao Deng, Xiaohua Yao. Well-posedness and ill-posedness for the 3D generalized Navier-Stokes equations in $\dot{F}^{-\alpha,r}_{\frac{3}{\alpha-1}}$. Discrete and Continuous Dynamical Systems, 2014, 34 (2) : 437-459. doi: 10.3934/dcds.2014.34.437

[2]

Kendry J. Vivas, Víctor F. Sirvent. Metric entropy for set-valued maps. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022010

[3]

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

[4]

Xing Wang, Nan-Jing Huang. Stability analysis for set-valued vector mixed variational inequalities in real reflexive Banach spaces. Journal of Industrial and Management Optimization, 2013, 9 (1) : 57-74. doi: 10.3934/jimo.2013.9.57

[5]

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

[6]

Qingbang Zhang, Caozong Cheng, Xuanxuan Li. Generalized minimax theorems for two set-valued mappings. Journal of Industrial and Management Optimization, 2013, 9 (1) : 1-12. doi: 10.3934/jimo.2013.9.1

[7]

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

[8]

Shay Kels, Nira Dyn. Bernstein-type approximation of set-valued functions in the symmetric difference metric. Discrete and Continuous Dynamical Systems, 2014, 34 (3) : 1041-1060. doi: 10.3934/dcds.2014.34.1041

[9]

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

[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]

Roger Metzger, Carlos Arnoldo Morales Rojas, Phillipe Thieullen. Topological stability in set-valued dynamics. Discrete and Continuous Dynamical Systems - B, 2017, 22 (5) : 1965-1975. doi: 10.3934/dcdsb.2017115

[12]

Geng-Hua Li, Sheng-Jie Li. Unified optimality conditions for set-valued optimizations. Journal of Industrial and Management Optimization, 2019, 15 (3) : 1101-1116. doi: 10.3934/jimo.2018087

[13]

Dante Carrasco-Olivera, Roger Metzger Alvan, Carlos Arnoldo Morales Rojas. Topological entropy for set-valued maps. Discrete and Continuous Dynamical Systems - B, 2015, 20 (10) : 3461-3474. doi: 10.3934/dcdsb.2015.20.3461

[14]

Yiming Ding. Renormalization and $\alpha$-limit set for expanding Lorenz maps. Discrete and Continuous Dynamical Systems, 2011, 29 (3) : 979-999. doi: 10.3934/dcds.2011.29.979

[15]

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

[16]

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

[17]

Yu Zhang, Tao Chen. Minimax problems for set-valued mappings with set optimization. Numerical Algebra, Control and Optimization, 2014, 4 (4) : 327-340. doi: 10.3934/naco.2014.4.327

[18]

Shouming Zhou, Chunlai Mu, Liangchen Wang. Well-posedness, blow-up phenomena and global existence for the generalized $b$-equation with higher-order nonlinearities and weak dissipation. Discrete and Continuous Dynamical Systems, 2014, 34 (2) : 843-867. doi: 10.3934/dcds.2014.34.843

[19]

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

[20]

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

 Impact Factor: 

Metrics

  • PDF downloads (38)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]