Abstract
This paper is devoted to the Levitin–Polyak well-posedness by perturbations for a class of general systems of set-valued vector quasi-equilibrium problems (SSVQEP) in Hausdorff topological vector spaces. Existence of solution for the system of set-valued vector quasi-equilibrium problem with respect to a parameter (PSSVQEP) and its dual problem are established. Some sufficient and necessary conditions for the Levitin–Polyak well-posedness by perturbations are derived by the method of continuous selection. We also explore the relationships among these Levitin–Polyak well-posedness by perturbations, the existence and uniqueness of solution to (SSVQEP). By virtue of the nonlinear scalarization technique, a parametric gap function g for (PSSVQEP) is introduced, which is distinct from that of Peng (J Glob Optim 52:779–795, 2012). The continuity of the parametric gap function g is proved. Finally, the relations between these Levitin–Polyak well-posedness by perturbations of (SSVQEP) and that of a corresponding minimization problem with functional constraints are also established under quite mild assumptions.
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References
Ansari QH, Schaible S, Yao JC (2000) System of vector equilibrium problems and its applications. J Optim Theory Appl 107: 547–557
Ansari QH, Schaible S, Yao JC (2002) The system of generalized vector equilibrium problems with applications. J Glob Optim 22: 3–16
Aubin JP, Ekeland I (1984) Applied nonlinear analysis. Wiley, New York
Berge C (1963) Topological spaces. Oliver and Boyd, London
Ceng LC, Hadjisavvas N, Schaible S, Yao JC (2008) Well-posedness for mixed quasivariational-like inequalities. J Optim Theory Appl 139: 109–125
Chen Z (1988) Continuous selections of set-valued mappings and fixed point theorems. Acta Math Sinica (Chinese) 31: 456–463
Chen GY, Yang XQ (2002) Characterizations of variable domination structures via nonlinear scalarization. J Optim Theory Appl 112: 97–110
Chen JW, Wan Z (2011) Existence of solutions and convergence analysis for a system of quasivariational inclusions in Banach spaces. J Inequal Appl 2011: 1–18
Chen GY, Goh CJ, Yang XQ (1999) Vector network equilibrium problems and nonlinear scalarization methods. Math Meth Oper Res 49: 239–253
Chen GY, Goh CJ, Yang XQ (2000) On gap functions for vector variational inequalities. In: Giannessi F. (ed) Vector variational inequalities and vector equilibria. Kluwer Academic, Dordrecht, pp 55–72
Chen GY, Yang XQ, Yu H (2005a) A nonlinear scalarization function and generalized quasi-vector equilibrium problems. J Glob Optim 32: 451–466
Chen GY, Huang XX, Yang XQ (2005b) Vector optimization: set-valued and variational analysis. In: Lecture notes in economics and mathematical systems, vol 285. Springer, Berlin, pp 408–416
Chen JW, Cho YJ, Wan Z (2011) Shrinking projection algorithms for equilibrium problems with a bifunction defined on the dual space of a Banach space. Fixed Point Theory Appl 2011: 91
Chen JW, Wan Z, Zou YZ (2012) Strong convergence theorems for firmly nonexpansive-type mappings and equilibrium problems in Banach spaces. Optimization doi:10.1080/02331934.2011.626779
Ding XP, Kim WK, Tan KK (1992) A selection theorem an its applications. Bull Austral Math Soc 46: 205–212
Fang M, Huang NJ, Kim J (2006) Existence results for systems of vector equilibrium problems. J Glob Optim 35: 71–83
Fang YP, Hu R, Huang NJ (2008) Well-posedness for equilibrium problems and for optimization problems with equilibrium constraints. Comput Math Appl 55: 89–100
Fang YP, Huang NJ, Yao JC (2010) Well-posedness by perturbations of mixed variational inequalities in Banach spaces. Eur J Oper Res 201: 682–692
Fu JY (2005) Vector equilibrium problems existence theorems and convexity of solution set. J Glob Optim 31: 109–119
Furi M, Vignoli A (1970) About well-posed optimization problems for functions in metric spaces. J Optim Theory Appl 5: 225–229
Giannessi F (1998) On Minty variational principle. In: Giannessi F, Komloski S, Tapcsack T (eds) New trends in mathematical programming. Kluwer Academic Publisher, Dordrech, pp 93–99
Giannessi F (2000) Vector variational inequalities and vector equilibria: mathematical theories. Kluwer Academic, Dordrecht
Gutev VG (1998) Continuous selections for continuous set-valued mappings and finite-dimensional sets. Set Valued Anal 6: 149–170
Homidan SA, Ansari QH, Schaible S (2007) Existence of solutions of systems of generalized implicit vector variational inequalities. J Optim Theory Appl 134: 515–531
Huang NJ, Li J, Yao JC (2007) Gap functions and existence of solutions for a system of vector equilibrium problems. J Optim Theory Appl 133: 201–212
Huang XX, Yang XQ (2006) Generalized Levitin–Polyak well-posedness in constrained optimization. SIAM J Optim 17: 243–258
Huang XX, Yang XQ (2007) Levitin–Polyak well-posedness in generalized variational inequalities problems with functional constraints. J Ind Manag Optim 3: 671–684
Huang XX, Yang XQ (2010) Levitin–Polyak well-posedness of vector variational inequality problems with functional constraints. Numer Funct Anal Optim 31: 671–684
Huang XX, Yang XQ, Zhu DL (2009) Levitin–Polyak well-posedness of variational inequality problems with functional constraints. J Glob Optim 2: 159–174
Hu R, Fang YP, Huang NJ (2010a) Levitin–Polyak well-posedness for variational inequalities and for optimization problems with variational inequalities. J Ind Manag Optim 6: 465–481
Hu R, Fang YP, Huang NJ, Wong MM (2010b) Well-posedness of systems of equilibrium problems. Taiwan J Math 14: 2435–2446
Jiang B, Zhang J, Huang XX (2009) Levitin–Polyak well-posedness of generalized quasivariational inequalities with functional constraints. Nonlinear Anal 70: 1492–1530
Konsulova AS, Revalski JP (1994) Constrained convex optimization problems well-posedness and stability. Numer Funct Anal Optim 7(8): 889–907
Kuratowski K (1968) Topology, vol. 1 and 2. Academic Press, New York
Lalitha CS, Bhatia G (2010) Well-posedness for parametric quasivariational inequality problems and for optimizations problems with quasivariational inequality constraints. Optimization 59: 997–1011
Lemaire B, Ould Ahmed Salem C, Revalski JP (2002) Well-posedness by perturbations of variational inequalities. J Optim Theory Appl 115: 345–368
Levitin ES, Polyak BT (1966) Convergence of minimizing sequences in conditional extremum problems. Soviet Math Dokl 7: 764–767
Lignola MB, Morgan J (2000) Well-posedness for optimization problems with constraints defined by variational inequalities having a unique solution. J Glob Optim 16: 57–67
Lignola MB, Morgan J (2001) Approximating solutions and α- well-posedness for variational inequalities and Nash equilibria. In: Decision and control in management science, Kluwer Academic Publishers, pp 367–378
Li J, He Z (2005) Gap functions and existence of solutions to generalized vector variational inequalities. Appl Math Lett 18: 989–1000
Li SJ, Li MH (2009) Levitin–Polyak well-posedness of vector equilibrium problems. Math Meth Oper Res 69: 125–140
Li SJ, Teo KL, Yang XQ, Wu SY (2006) Gap functions and existence of solutions to generalized vector quasi-equilibrium problems. J Glob Optim 34: 427–440
Lucchetti R, Patrone F (1981) A characterization of Tykhonov well-posedness for minimum problems with applications to variational inequalities. Numer Funct Anal Optim 3: 461–476
Peng JW, Wu SY (2010) The generalized Tykhonov well-posedness for system of vector quasi-equilibrium problems. Optim Lett 4: 501–512
Peng JW, Wang Y, Zhao LJ (2009) Generalized Levitin–Polyak well-posedness of vector equilibrium problems. Fixed Point Theory Appl 2009: 1–14
Peng JW, Wu SY, Wang Y (2012) Levitin–Polyak well-posedness of generalized vector quasi-equilibrium problems with functional constraints. J Glob Optim 52: 779–795
Tykhonov AN (1966) On the stability of the functional optimization problem. USSR J Comput Math Math Phys 6: 631–634
Xu Z, Zhu DL, Huang XX (2008) Levitin–Polyak well-posedness in generalized vector variational inequality problem with functional constraints. Math Meth Oper Res 67: 505–524
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Chen, JW., Wan, Z. & Cho, Y.J. Levitin–Polyak well-posedness by perturbations for systems of set-valued vector quasi-equilibrium problems. Math Meth Oper Res 77, 33–64 (2013). https://doi.org/10.1007/s00186-012-0414-5
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DOI: https://doi.org/10.1007/s00186-012-0414-5
Keywords
- System of set-valued vector quasi-equilibrium problem
- Existence theorem
- Levitin–Polyak well-posedness by perturbations
- Parametric gap function
- Nonlinear scalarization function