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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.

[2]

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

[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.

[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.

[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.

[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.

[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.

[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.

[13]

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

[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.

[15]

K. Kuratowski, "Topology,", (Vols. 1 and 2), (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. 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. 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:, (2001), 367.

[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.

[20]

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

[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.

[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.

[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.

[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.

[25]

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

[28]

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

[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.

[30]

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

[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.

[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.

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.

[2]

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

[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.

[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.

[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.

[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.

[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.

[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.

[13]

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

[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.

[15]

K. Kuratowski, "Topology,", (Vols. 1 and 2), (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. 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. 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:, (2001), 367.

[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.

[20]

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

[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.

[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.

[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.

[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.

[25]

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

[28]

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

[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.

[30]

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

[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.

[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.

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