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

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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

Jiawei Chen ^1, , Zhongping Wan ^2, and Liuyang Yuan ^2,

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

Abstract
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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.

Keywords: existence of solution, (generalized) $\alpha$-well-posedness, metric characterization, System of constrained set-valued variational inequalities, $\alpha$-approximating sequence..

Mathematics Subject Classification: Primary: 47J04, 49K40; Secondary: 90C3.

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.

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