A NEW CLASS OF BILEVEL GENERALIZED MIXED EQUILIBRIUM PROBLEMS IN BANACH SPACES

doi:10.1016/S0252-9602(12)60124-6

Abstract

Abstract:

A new class of bilevel generalized mixed equilibrium problems involving set-valued mappings is introduced and studied in a real Banach space. By using the auxiliary principle technique, new iterative algorithms for solving the generalized mixed equilibrium problems and bilevel generalized mixed equilibrium problems involving set-valued map-pings are suggested and analyzed. Existence of solutions and strong convergence of the iterative sequences generated by the algorithms are proved under quite mild conditions. The behavior of the solution set of the generalized mixed equilibrium problems and bilevel generalized mixed equilibrium problems is also discussed. These results are new and gen-eralize some recent results in this field.

Key words: generalized mixed equilibrium problem, bilevel generalized mixed equilib-rium problem involving set-valued mappings, monotonicity, auxiliary prin-ciple, iterative algorithm, Banach space

CLC Number:

47J22

DING Xie-Ping. A NEW CLASS OF BILEVEL GENERALIZED MIXED EQUILIBRIUM PROBLEMS IN BANACH SPACES[J].Acta mathematica scientia,Series B, 2012, 32(4): 1571-1583.

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References

[1] Liou Y C, Yao J C. Bilevel decision via variational inequalities. Comput Math Appl, 2005, 49: 1243–1253

[2] Ding X P, Liou Y C. Bilevel optimization problems in topological spaces. Taiwanese J Math, 2006, 10(1): 173–179

[3] Birbil S, Bouza G, Frenk J B G, Still G. Equilibrium constraint optimization problems. Eur J Operation Res, 2006, 169: 1108–1127

[4] Lin L J. Mathematical programming with system of equilibrium constraints. J Glob Optim, 2007, 37(2): 195–213

[5] Lin L J. Existence theorems for bilevel problem with applications to mathematical program with equilib-rium constraint and semi-infinite problem. J Optim Theory Appl, 2008, 137(1): 27–40

[6] Lin L J, Shie H J. Existence theorems of quasivariational inclusion problems with applications to bilevel problems and mathematical problems with equilibrium constraint. J Optim Theory Appl, 2008, 138(3): 445–457

[7] Ding X P. System of generalized vector quasi-variational inclusions and generalized vector quasi-optimizations in locally FC-uniform spaces. Appl Math Mech, 2009, 30(3): 263–274

[8] Ding X P. Mathematical programs with system of generalized vector quasi-equilibrium constraints in FC-spaces. Acta Math Sci, 2010, 30B(4): 1257–1268

[9] Moudafi A. Proximal methods for a class of bilevel monotone equilibrium problems. J Glob Optim, 2010, 47(2): 287–292

[10] Ding X P. Auxiliary principle and algorithm for mixed equilibrium problems and bilevel mixed equilibrium problems in Banach spaces. J Optim Theory Appl, 2010, 146(2): 347–357

[11] Ding X P. Iterative algorithm of solutions for generalized mixed implicit equilibrium-like problems. Appl Math Comput, 2005, 162(2): 799–809

[12] Ding X P, Lin Y C, Yao J C. Predictor-corrector algorithms for solving generalized mixed implicit quasi-equilibrium problems. Appl Math Mech, 2006, 27(9): 1157–1164

[13] Moudafi A. Mixed equilibrium problems: sensitivity analysis and algorithmic aspects. Comput Math Appl, 2002, 44: 1099–1108

[14] Huang N J, Lan H Y, Cho Y J. Sensitivity analysis for nonlinear generalized mixed implicit equilibrium problems with non-monotone set-valued mappings. J Comput Appl Math, 2006, 196: 608–618

[15] Kazmi K R, Khan F A. Existence and iterative approximation of solutions of generalized mixed equilibrium problems. Comput Math Appl, 2008, 56: 1314–1321

[16] Ding X P. Existence and algorithm of solutions for nonlinear mixed quasi-variational inequalities in Banach spaces. J Comput Appl Math, 2003, 157: 419–434

[17] Ding X P. Existence and algorithm of solutions for mixed variational-like inequalities in Banach spaces. J Optim Theory Appl, 2005, 127(2): 285–302

[18] Ding X P, Yao J C. Existence and algorithm of solutions for mixed quasi-variational-like inclusions in Banach spaces. Comput Math Appl, 2005, 49(5/6): 857–869

[19] Ding X P, Yao J C, Zeng L C. Existence and algorithm of solutions for generalized strongly nonlinear mixed variational-like inequalities in Banach spaces. Comput Math Appl, 2008, 55: 669–679

[20] Zeng L C, Schaible S, Yao J C. Iterative algorithm for generalized set-valued strongly nonlinear mixed variational-like inequalities. J Optim Theory Appl, 2005, 124(3): 725–738

[21] Kazmi K R, Khan F A. Auxialiary problems and algorithm for a system of generalized variational-like inequality problems. Appl Math Comput, 2007, 187: 789–796

[22] Antipin A S. Iterative gradient prediction-type methods for computing fixed-point of extremal map-pings//Guddat H, Jonden H Th. Nizicka F, Still G, Twitt F, eds. Parametric Optimization and Related Topics IV. Peter Lang, Frankfurt Main, 1997: 11–24

[23] Ding X P, Tan K K. A minimax inequality with applications to existence of equilibrium point and fixed point theorems. Colloq Math, 1992, 63: 233–247

[24] Chang S S. Variational Inequalities and Complementary Problems: Theory and Applications. Shanghai: Shanghai Scientific and Technical Press, 1991 (in Chinese)

[25] Nadler S B. Set-valued contractive mappings. Pacfic J Math, 1969, 30: 475–485