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An integrated approach based on Fuzzy Inference System for scheduling and process planning through multiple objectives
Error bounds of regularized gap functions for nonmonotone Ky Fan inequalities
1. | School of Mathematics and Finance, Chongqing University of Arts and Sciences, Yongchuan, Chongqing, 402160, China |
2. | College of Mathematics and Statistics, Chongqing University, Chongqing, 401331, China |
In this paper, the Clarke generalized Jacobian of the generalized regularized gap function for a nonmonotone Ky Fan inequality is studied. Then, based on the Clarke generalized Jacobian, we derive a global error bound for the nonmonotone Ky Fan inequalities. Finally, an application is given to provide a descent method.
References:
[1] |
G. Auchmuty,
Variational principles for variational inequalities, Numer. Funct. Anal. Optim., 10 (1989), 863-874.
doi: 10.1080/01630568908816335. |
[2] |
G. Bigi, M. Castellani and M. Pappalardo,
A new solution method for equilibrium problems, Optim. Methods Softw., 24 (2009), 895-911.
doi: 10.1080/10556780902855620. |
[3] |
G. Bigi, M. Castellani, M. Pappalardo and M. Passacantando,
Existence and solution methods for equilibria, European J. Oper. Res., 227 (2013), 1-11.
doi: 10.1016/j.ejor.2012.11.037. |
[4] |
E. Blum and W. Oettli,
From optimization and variational inequalities to equilibrium problems, Math. Student, 63 (1994), 123-145.
|
[5] |
O. Chadli, I. V. Konnov and J. C. Yao,
Descent methods for equilibrium problems in a Banach space, Comput. Math. Appl., 48 (2004), 609-616.
doi: 10.1016/j.camwa.2003.05.011. |
[6] |
O. Chadli and S. Schaible,
Regularized equilibrium problems with application to noncoercive hemivariational inequalities, J. Optim. Theory Appl., 121 (2004), 571-596.
doi: 10.1023/B:JOTA.0000037604.96151.26. |
[7] |
O. Chadli, Z. H. Liu and J. C. Yao,
Applications of equilibrium problems to a class of noncoercive variational inequalities, J. Optim. Theory Appl., 132 (2007), 89-110.
doi: 10.1007/s10957-006-9072-1. |
[8] |
C. Charitha,
A note on D-gap functions for equilibrium problems, Optimization, 62 (2013), 211-226.
doi: 10.1080/02331934.2011.583987. |
[9] |
F. H. Clarke, Optimization and Nonsmooth Analysis, Wiley, New York, 1983. |
[10] |
F. Facchinei and J. S. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems, Springer, Berlin Helidelberg, New York, 2003. |
[11] |
K. Fan, A minimax inequality and applications: Inequality Ⅲ, Proc. Third Sympos., Univ. California, Los Angeles, Calif., 1969; dedicated to the memory of Theodore S. Motzkin, Academic Press, New York, (1972), 103–113. |
[12] |
M. Fukushima,
Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems, Math. Program., 53 (1992), 99-110.
doi: 10.1007/BF01585696. |
[13] |
F. Giannessi, A. Maugeri and P. M. Pardalos, Equilibrium Problems: Nonsmooth Optimization and Variational Inequality Models, Nonconvex Optimization and its Applications, 58. Kluwer Academic Publishers, Dordrecht, 2001.
doi: 10.1007/0-306-48026-3_12. |
[14] |
L. R. Huang and K. F. Ng,
Equivalent optimization formulations and error bounds for variational inequality problems, J. Optim. Theory Appl., 125 (2005), 299-314.
doi: 10.1007/s10957-004-1839-7. |
[15] |
A. N. Iusem and W. Sosa,
New existence results for equilibrium problems, Nonlinear Anal., 52 (2003), 621-635.
doi: 10.1016/S0362-546X(02)00154-2. |
[16] |
H. Y. Jiang and L. Q. Qi,
Local uniqueness and convergence of iterative methods for nonsmooth variational inequalities, J. Math. Anal. Appl., 196 (1995), 314-331.
doi: 10.1006/jmaa.1995.1412. |
[17] |
I. V. Konnov and M. S. S. Ali,
Descent methods for monotone equilibrium problems in Banach spaces, J. Comput. Appl. Math., 188 (2006), 165-179.
doi: 10.1016/j.cam.2005.04.004. |
[18] |
I. V. Konnov and O. V. Pinyagina,
D-gap functions for a class of equilibrium problems in Banach spaces, Comput. Methods Appl. Math., 3 (2003), 274-286.
doi: 10.2478/cmam-2003-0018. |
[19] |
I. V. Konnov,
Combined relaxation method for monotone equilibrium problems, J. Optim. Theory Appl., 111 (2001), 327-340.
doi: 10.1023/A:1011930301552. |
[20] |
Y. S. Ledyaev, R. J. Stern and P. R. Wolenski, Nonsmooth Analysis and Control Theory, Graduate Texts in Mathematics. Springer-Verlag, New York, 1998. |
[21] |
G. Li and K. F. Ng,
Error bounds of generalized D-gap functions for nonsmooth and nonmonotone variational inequality problems, SIAM J. Optim., 20 (2009), 667-690.
doi: 10.1137/070696283. |
[22] |
G. Li, C. Tang and Z. Wei,
Error bound results for generalized D-gap functions of nonsmooth variational inequality problems, J. Comput. Appl. Math., 233 (2010), 2795-2806.
doi: 10.1016/j.cam.2009.11.025. |
[23] |
G. Mastroeni, On auxiliary principle for equilibrium problems, In: Daniele, P., Giannessi, F., Maugeri, A. (eds.): Equilibrium Problems and Variational Models, Kluwer Academic Publishers, Dordrecht, 68(2003), 289-298.
doi: 10.1007/978-1-4613-0239-1_15. |
[24] |
G. Mastroeni,
Gap functions for equilibrium problems, J. Glob. Optim., 27 (2003), 411-426.
doi: 10.1023/A:1026050425030. |
[25] |
L. D. Muu and T. D. Quoc,
Regularization algorithms for solving monotone Ky Fan inequalities with application to a Nash-Cournot equilibrium model, J. Optim. Theory Appl., 142 (2009), 185-204.
doi: 10.1007/s10957-009-9529-0. |
[26] |
M. A. Noor and W. Oettli,
On general nonlinear complementarity problems and quasi-equilibria, Le Matematiche (Catania), 49 (1994), 313-331.
|
[27] |
M. A. Noor,
Auxiliary principle technique for equilibrium problems, J. Optim. Theory Appl., 122 (2004), 371-386.
doi: 10.1023/B:JOTA.0000042526.24671.b2. |
[28] |
J. M. Peng,
Equivalence of variational inequality problems to unconstrained optimization, Math. Program., 78 (1997), 347-355.
doi: 10.1007/BF02614360. |
[29] |
H. Rademacher,
Über partielle und totale differenzierbarkeit von Funktionen mehrerer Variabeln und Über die Transformation der Doppelintegrale, Math. Ann., 79 (1919), 340-359.
doi: 10.1007/BF01498415. |
[30] |
L. C. Zeng and J. C. Yao,
Modified combined relaxation method for general monotone equilibrium problems in Hilbert spaces, J. Optim. Theory Appl., 131 (2006), 469-483.
doi: 10.1007/s10957-006-9162-0. |
[31] |
L. P. Zhang and J. Y. Han,
Unconstrained optimization reformulations of equilibrium problems, Acta Math. Sin. (Engl. Ser.), 25 (2009), 343-354.
doi: 10.1007/s10114-008-7096-1. |
[32] |
L. P. Zhang and S. Y. Wu,
An algorithm based on the generalized D-gap function for equilibrium problems, J. Comput. Appl. Math., 231 (2009), 403-411.
doi: 10.1016/j.cam.2009.03.006. |
show all references
References:
[1] |
G. Auchmuty,
Variational principles for variational inequalities, Numer. Funct. Anal. Optim., 10 (1989), 863-874.
doi: 10.1080/01630568908816335. |
[2] |
G. Bigi, M. Castellani and M. Pappalardo,
A new solution method for equilibrium problems, Optim. Methods Softw., 24 (2009), 895-911.
doi: 10.1080/10556780902855620. |
[3] |
G. Bigi, M. Castellani, M. Pappalardo and M. Passacantando,
Existence and solution methods for equilibria, European J. Oper. Res., 227 (2013), 1-11.
doi: 10.1016/j.ejor.2012.11.037. |
[4] |
E. Blum and W. Oettli,
From optimization and variational inequalities to equilibrium problems, Math. Student, 63 (1994), 123-145.
|
[5] |
O. Chadli, I. V. Konnov and J. C. Yao,
Descent methods for equilibrium problems in a Banach space, Comput. Math. Appl., 48 (2004), 609-616.
doi: 10.1016/j.camwa.2003.05.011. |
[6] |
O. Chadli and S. Schaible,
Regularized equilibrium problems with application to noncoercive hemivariational inequalities, J. Optim. Theory Appl., 121 (2004), 571-596.
doi: 10.1023/B:JOTA.0000037604.96151.26. |
[7] |
O. Chadli, Z. H. Liu and J. C. Yao,
Applications of equilibrium problems to a class of noncoercive variational inequalities, J. Optim. Theory Appl., 132 (2007), 89-110.
doi: 10.1007/s10957-006-9072-1. |
[8] |
C. Charitha,
A note on D-gap functions for equilibrium problems, Optimization, 62 (2013), 211-226.
doi: 10.1080/02331934.2011.583987. |
[9] |
F. H. Clarke, Optimization and Nonsmooth Analysis, Wiley, New York, 1983. |
[10] |
F. Facchinei and J. S. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems, Springer, Berlin Helidelberg, New York, 2003. |
[11] |
K. Fan, A minimax inequality and applications: Inequality Ⅲ, Proc. Third Sympos., Univ. California, Los Angeles, Calif., 1969; dedicated to the memory of Theodore S. Motzkin, Academic Press, New York, (1972), 103–113. |
[12] |
M. Fukushima,
Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems, Math. Program., 53 (1992), 99-110.
doi: 10.1007/BF01585696. |
[13] |
F. Giannessi, A. Maugeri and P. M. Pardalos, Equilibrium Problems: Nonsmooth Optimization and Variational Inequality Models, Nonconvex Optimization and its Applications, 58. Kluwer Academic Publishers, Dordrecht, 2001.
doi: 10.1007/0-306-48026-3_12. |
[14] |
L. R. Huang and K. F. Ng,
Equivalent optimization formulations and error bounds for variational inequality problems, J. Optim. Theory Appl., 125 (2005), 299-314.
doi: 10.1007/s10957-004-1839-7. |
[15] |
A. N. Iusem and W. Sosa,
New existence results for equilibrium problems, Nonlinear Anal., 52 (2003), 621-635.
doi: 10.1016/S0362-546X(02)00154-2. |
[16] |
H. Y. Jiang and L. Q. Qi,
Local uniqueness and convergence of iterative methods for nonsmooth variational inequalities, J. Math. Anal. Appl., 196 (1995), 314-331.
doi: 10.1006/jmaa.1995.1412. |
[17] |
I. V. Konnov and M. S. S. Ali,
Descent methods for monotone equilibrium problems in Banach spaces, J. Comput. Appl. Math., 188 (2006), 165-179.
doi: 10.1016/j.cam.2005.04.004. |
[18] |
I. V. Konnov and O. V. Pinyagina,
D-gap functions for a class of equilibrium problems in Banach spaces, Comput. Methods Appl. Math., 3 (2003), 274-286.
doi: 10.2478/cmam-2003-0018. |
[19] |
I. V. Konnov,
Combined relaxation method for monotone equilibrium problems, J. Optim. Theory Appl., 111 (2001), 327-340.
doi: 10.1023/A:1011930301552. |
[20] |
Y. S. Ledyaev, R. J. Stern and P. R. Wolenski, Nonsmooth Analysis and Control Theory, Graduate Texts in Mathematics. Springer-Verlag, New York, 1998. |
[21] |
G. Li and K. F. Ng,
Error bounds of generalized D-gap functions for nonsmooth and nonmonotone variational inequality problems, SIAM J. Optim., 20 (2009), 667-690.
doi: 10.1137/070696283. |
[22] |
G. Li, C. Tang and Z. Wei,
Error bound results for generalized D-gap functions of nonsmooth variational inequality problems, J. Comput. Appl. Math., 233 (2010), 2795-2806.
doi: 10.1016/j.cam.2009.11.025. |
[23] |
G. Mastroeni, On auxiliary principle for equilibrium problems, In: Daniele, P., Giannessi, F., Maugeri, A. (eds.): Equilibrium Problems and Variational Models, Kluwer Academic Publishers, Dordrecht, 68(2003), 289-298.
doi: 10.1007/978-1-4613-0239-1_15. |
[24] |
G. Mastroeni,
Gap functions for equilibrium problems, J. Glob. Optim., 27 (2003), 411-426.
doi: 10.1023/A:1026050425030. |
[25] |
L. D. Muu and T. D. Quoc,
Regularization algorithms for solving monotone Ky Fan inequalities with application to a Nash-Cournot equilibrium model, J. Optim. Theory Appl., 142 (2009), 185-204.
doi: 10.1007/s10957-009-9529-0. |
[26] |
M. A. Noor and W. Oettli,
On general nonlinear complementarity problems and quasi-equilibria, Le Matematiche (Catania), 49 (1994), 313-331.
|
[27] |
M. A. Noor,
Auxiliary principle technique for equilibrium problems, J. Optim. Theory Appl., 122 (2004), 371-386.
doi: 10.1023/B:JOTA.0000042526.24671.b2. |
[28] |
J. M. Peng,
Equivalence of variational inequality problems to unconstrained optimization, Math. Program., 78 (1997), 347-355.
doi: 10.1007/BF02614360. |
[29] |
H. Rademacher,
Über partielle und totale differenzierbarkeit von Funktionen mehrerer Variabeln und Über die Transformation der Doppelintegrale, Math. Ann., 79 (1919), 340-359.
doi: 10.1007/BF01498415. |
[30] |
L. C. Zeng and J. C. Yao,
Modified combined relaxation method for general monotone equilibrium problems in Hilbert spaces, J. Optim. Theory Appl., 131 (2006), 469-483.
doi: 10.1007/s10957-006-9162-0. |
[31] |
L. P. Zhang and J. Y. Han,
Unconstrained optimization reformulations of equilibrium problems, Acta Math. Sin. (Engl. Ser.), 25 (2009), 343-354.
doi: 10.1007/s10114-008-7096-1. |
[32] |
L. P. Zhang and S. Y. Wu,
An algorithm based on the generalized D-gap function for equilibrium problems, J. Comput. Appl. Math., 231 (2009), 403-411.
doi: 10.1016/j.cam.2009.03.006. |
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