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May  2020, 16(3): 1261-1272. doi: 10.3934/jimo.2019001

Error bounds of regularized gap functions for nonmonotone Ky Fan inequalities

Minghua Li 1,, , Chunrong Chen 2, and  Shengjie Li 2,

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

* Corresponding author: Minghua Li

Received  November 2016 Revised  November 2018 Published  May 2020 Early access  March 2019

Fund Project: The work was supported in part by the National Natural Science Foundation of China (Grant numbers: 11301418, 11301567, 11571055), the Natural Science Foundation of Chongqing Municipal Science and Technology Commission (Grant numbers: cstc2016jcyjA0141, cstc2016jcyjA0270, cstc2018jcyjAX0226), the Basic Science and Frontier Technology Research of Yongchuan (Grant number: Ycstc, 2018nb1401), the Fundamental Research Funds for the Central Universities (Grant Number: 106112017CDJZRPY0020), the Foundation for High-level Talents of Chongqing University of Art and Sciences (Grant numbers: R2016SC13, P2017SC01), the Chongqing Key Laboratory of Group and Graph Theories and Applications and the Key Laboratory of Complex Data Analysis and Artificial Intelligence of Chongqing Municipal Science and Technology Commission

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.

Keywords: Error bound, regularized gap function, Ky Fan inequality, descent method.
Mathematics Subject Classification: Primary: 65K10, 90C31; Secondary: 90C33.
Citation: Minghua Li, Chunrong Chen, Shengjie Li. Error bounds of regularized gap functions for nonmonotone Ky Fan inequalities. Journal of Industrial and Management Optimization, 2020, 16 (3) : 1261-1272. doi: 10.3934/jimo.2019001
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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