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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 & 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.   Google Scholar

[2]

H. Attouch, "E.D.P.associées à de sous-différentiels,", Thèse de Doctorat d'état ES Sciences Mathématiques, (1976).   Google Scholar

[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.  doi: 10.1007/s10957-008-9428-9.  Google Scholar

[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.  doi: 10.1007/s00186-012-0414-5.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.1016/j.camwa.2007.03.019.  Google Scholar

[8]

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

[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.  doi: 10.3934/jimo.2007.3.671.  Google Scholar

[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.  doi: 10.1080/01630563.2010.485296.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.3934/jimo.2010.6.465.  Google Scholar

[13]

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

[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.   Google Scholar

[15]

K. Kuratowski, "Topology,", (Vols. 1 and 2), (1968).   Google Scholar

[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.  doi: 10.1080/02331930902878358.  Google Scholar

[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.  doi: 10.1023/A:1008370910807.  Google Scholar

[18]

M. B. Lignola and J. Morgan, Approximating solutions and $\alpha$-well-posedness for variational inequalities and Nash equilibria,, in:, (2001), 367.   Google Scholar

[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.  doi: 10.1007/s10898-006-9020-5.  Google Scholar

[20]

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

[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.   Google Scholar

[22]

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

[23]

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

[24]

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

[25]

D. Pascali and S. Sburlan, "Nonlinear Mappings of Monotone Type,", Martinus Nijhoff, (1978).   Google Scholar

[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.  doi: 10.1007/s11590-010-0179-9.  Google Scholar

[27]

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

[28]

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

[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.  doi: 10.1155/2010/614276.  Google Scholar

[30]

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

[31]

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

[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.  doi: 10.3934/naco.2011.1.261.  Google Scholar

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.   Google Scholar

[2]

H. Attouch, "E.D.P.associées à de sous-différentiels,", Thèse de Doctorat d'état ES Sciences Mathématiques, (1976).   Google Scholar

[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.  doi: 10.1007/s10957-008-9428-9.  Google Scholar

[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.  doi: 10.1007/s00186-012-0414-5.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.1016/j.camwa.2007.03.019.  Google Scholar

[8]

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

[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.  doi: 10.3934/jimo.2007.3.671.  Google Scholar

[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.  doi: 10.1080/01630563.2010.485296.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.3934/jimo.2010.6.465.  Google Scholar

[13]

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

[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.   Google Scholar

[15]

K. Kuratowski, "Topology,", (Vols. 1 and 2), (1968).   Google Scholar

[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.  doi: 10.1080/02331930902878358.  Google Scholar

[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.  doi: 10.1023/A:1008370910807.  Google Scholar

[18]

M. B. Lignola and J. Morgan, Approximating solutions and $\alpha$-well-posedness for variational inequalities and Nash equilibria,, in:, (2001), 367.   Google Scholar

[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.  doi: 10.1007/s10898-006-9020-5.  Google Scholar

[20]

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

[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.   Google Scholar

[22]

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

[23]

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

[24]

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

[25]

D. Pascali and S. Sburlan, "Nonlinear Mappings of Monotone Type,", Martinus Nijhoff, (1978).   Google Scholar

[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.  doi: 10.1007/s11590-010-0179-9.  Google Scholar

[27]

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

[28]

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

[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.  doi: 10.1155/2010/614276.  Google Scholar

[30]

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

[31]

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

[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.  doi: 10.3934/naco.2011.1.261.  Google Scholar

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