March  2022, 12(1): 63-78. doi: 10.3934/naco.2021051

Weak convergence theorems for symmetric generalized hybrid mappings and equilibrium problems

1. 

Department of Applied Mathematics, Pukyong National University, Busan, 48513, Korea

2. 

Department of Scientific Fundamentals, Vietnam Trade Union University, Hanoi, Vietnam

3. 

Department of Mathematics, Faculty of Information Technology, Le Quy Don Technical University, Hanoi, Vietnam

*Corresponding author

Received  March 2020 Revised  October 2021 Published  March 2022 Early access  November 2021

Fund Project: This work was supported by the National Research Foundation of Korea (NRF) Grant funded by the Korean Government (NRF-2019R1A2C1008672)

In this paper, we introduce three new iterative methods for finding a common point of the set of fixed points of a symmetric generalized hybrid mapping and the set of solutions of an equilibrium problem in a real Hilbert space. Each method can be considered as an combination of Ishikawa's process with the proximal point algorithm, the extragradient algorithm with or without linesearch. Under certain conditions on parameters, the iteration sequences generated by the proposed methods are proved to be weakly convergent to a solution of the problem. These results extend the previous results given in the literature. A numerical example is also provided to illustrate the proposed algorithms.

Citation: Do Sang Kim, Nguyen Ngoc Hai, Bui Van Dinh. Weak convergence theorems for symmetric generalized hybrid mappings and equilibrium problems. Numerical Algebra, Control & Optimization, 2022, 12 (1) : 63-78. doi: 10.3934/naco.2021051
References:
[1]

P. N. Anh, A hybrid extragradient method extended to fixed point problems and equilibrium problems, Optim., 62 (2013), 271-283.  doi: 10.1080/02331934.2011.607497.  Google Scholar

[2]

G. BigiM. CastellaniM. Pappalardo and M. Passacantando, Existence and solution methods for equilibria, Eur. J. Oper. Res., 227 (2013), 1-11.  doi: 10.1016/j.ejor.2012.11.037.  Google Scholar

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E. Blum and W. Oettli, From optimization and variational inequalities to equilibrium problems, Math. Student, 63 (1994), 127-149.   Google Scholar

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P. L. Combettes and A. Hirstoaga, Equilibrium programming in Hilbert spaces, J. Nonlinear Convex Anal., 6 (2005), 117-136.   Google Scholar

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J. ContrerasM. Klusch and J. B. Krawczyk, Numerical solution to Nash-Cournot equilibria in coupled constraint electricity markets, IEEE Trans. Power Syst., 19 (2004), 195-206.   Google Scholar

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B. V. Dinh and D. S. Kim, Extragradient algorithms for equilibrium problems and symmetric generalized hybrid mappings, Optim. Lett., 11 (2016), 537-553.  doi: 10.1007/s11590-016-1025-5.  Google Scholar

[7]

B. V. Dinh and L. D. Muu, A projection algorithm for solving pseudomonotone equilibrium problems and it's application to a class of bilevel equilibria, Optim., 64 (2015), 559-575.  doi: 10.1080/02331934.2013.773329.  Google Scholar

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B. V. DinhP. G. Hung and L. D. Muu, Bilevel optimization as a regularization approach to pseudomonotone equilibrium problems, Numer. Funct. Anal. Optim., 35 (2014), 539-563.  doi: 10.1080/01630563.2013.813857.  Google Scholar

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B. V. Dinh and L. D. Muu, On penalty and gap function methods for bilevel equilibrium problems, J. Appl. Math., (2011) DOI: 10.1155/2011/646452. doi: 10.1155/2011/646452.  Google Scholar

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F. Facchinei and J. S. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems, Springer, New York, 2003.  Google Scholar

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A. Genel and J. Lindenstrass, An example concerning fixed points, Isarel J. Math., 22 (1975), 81-86.  doi: 10.1007/BF02757276.  Google Scholar

[12]

D. V. HieuL. D. Muu and P. K. Anh, Parallel hybrid extragradient methods for pseudomonotone equilibrium problems and nonexpansive mappings, Numer. Algor., 73 (2016), 197-217.  doi: 10.1007/s11075-015-0092-5.  Google Scholar

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M. HojoT. Suzuki and W. Takahashi, Fixed point theorems and convergence theorems for generalized hybrid non-self mappings in Hilbert spaces, J. Nonlinear Convex Anal., 14 (2013), 363-376.   Google Scholar

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H. Iiduka, Fixed point optimization algorithm and its application to power control in CDMA data networks, Math. Program., Ser. A, 133 (2012), 227-242.  doi: 10.1007/s10107-010-0427-x.  Google Scholar

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S. Ishikawa, Fixed points by a new iteration method, Proc. Amer. Math. Soc., 40 (1974), 147-150.  doi: 10.2307/2039245.  Google Scholar

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S. Itoh and W. Takahashi, The common fixed point theory of single-valued mappings and multi-valued mappings, Pacific J. Math., 79 (1978), 493-508.   Google Scholar

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T. Kawasaki and W. Takahashi, Existence and mean approximation of fixed points of generalized hybrid mappings in Hilbert spaces, J. Nonlinear Convex Anal., 14 (2013), 71-87.   Google Scholar

[18]

F. Kohsaka and W. Takahashi, Fixed point theorems for a class of nonlinear mappings related to maximal monotone operators in Banach spaces, Arch. Math., 91 (2008), 166-177.  doi: 10.1007/s00013-008-2545-8.  Google Scholar

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I. V. Konnov, Combined Relaxation Methods for Variational Inequalities, Lecture Notes in Economics and Mathematical System, 495 (2001), Springer, Berlin. doi: 10.1007/978-3-642-56886-2.  Google Scholar

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G. M. Korpelevich, The extragradient method for finding saddle points and other problems, Matekon, 12 (1976), 747-756.   Google Scholar

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W. R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc., 4 (1953), 506-510.  doi: 10.2307/2032162.  Google Scholar

[22]

F. Moradlou and Al izadeh, Strong convergence theorem by a new iterative method for equilibrium problems and symmetric generalized hybrid mappings, Mediterr. J. Math., 13 (2016), 379-390.  doi: 10.1007/s00009-014-0462-6.  Google Scholar

[23]

A. Moudafi, Proximal point algorithm extended to equilibrium problems, J. Nat. Geom., 15 (1999), 91-100.   Google Scholar

[24]

L. D. Muu and W. Oettli, Convergence of an adaptive penalty scheme for finding constrained equilibria, Nonlinear Anal.: TMA., 18 (1992), 1159-1166.  doi: 10.1016/0362-546X(92)90159-C.  Google Scholar

[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.  Google Scholar

[26] R. T. Rockafellar, Convex Analysis, Princeton University Press, 1970.   Google Scholar
[27]

J. Schu, Weak and strong convergence to fixed points of asymptotically noexpansive mappings, Bulletin of the Australian Math. Soc., 43 (1991), 153-159.  doi: 10.1017/S0004972700028884.  Google Scholar

[28]

A. Tada and W. Takahashi, Weak and strong convergence theorem for nonexpansive mapping and equilibrium problem, J. Optim. Theory Appl., 133 (2007), 359-370.  doi: 10.1007/s10957-007-9187-z.  Google Scholar

[29]

W. Takahashi and M. Toyoda, Weak convergence theorems for nonexpansive mappings and monotone mappings, J. Optim. Theory Appl., 118 (2003), 417-428.  doi: 10.1023/A:1025407607560.  Google Scholar

[30]

W. TakahashiN. C. Wong and J. C. Yao, Fixed point theorems for new generalized hybrid mappings in Hilbert spaces and applications, Taiwanese J. Math., 17 (2013), 1597-1611.  doi: 10.11650/tjm.17.2013.2921.  Google Scholar

[31]

W. Takahashi, Fixed point theorems for new nonlinear mappings in a Hilbert space, J. Nonlinear Convex Anal., 11 (2010), 79-88.   Google Scholar

[32]

D. Q. TranL. M. Dung and V. H. Nguyen, Extragradient algorithms extended to equilibrium problems, Optim., 57 (2008), 749-776.  doi: 10.1080/02331930601122876.  Google Scholar

[33]

N. N. TamJ. C. Yao and N. D. Yen, Solution methods for pseudomonotone variational inequalities, J. Optim. Theory Appl., 38 (2008), 253-273.  doi: 10.1007/s10957-008-9376-4.  Google Scholar

[34]

P. T. VuongJ. J. Strodiot and V. H. Nguyen, Extragradient methods and linesearch algorithms for solving Ky Fan inequalities and fixed point problems, J. Optim. Theory Appl., 155 (2013), 605-627.  doi: 10.1007/s10957-012-0085-7.  Google Scholar

[35]

H. K. Xu, A variable Krasnosel'ski$\mathop {\rm{i}}\limits^ \vee$-Mann algorithm and the multiple-set split feasibility problem, Inverse Problems, 22 (2006), 2021-2034.  doi: 10.1088/0266-5611/22/6/007.  Google Scholar

[36]

C. M. Yanes and H. K. Xu, Strong convergence of the $C Q$ method for fixed point iteration processes, Nonlinear Anal. TMA., 64 (2006), 2400-2411.  doi: 10.1016/j.na.2005.08.018.  Google Scholar

[37]

E. Zeidler, Nonlinear Functional Analysis and Its Applications I, Springer-Verlag, New York, 1986. doi: 10.1007/978-1-4612-4838-5.  Google Scholar

show all references

References:
[1]

P. N. Anh, A hybrid extragradient method extended to fixed point problems and equilibrium problems, Optim., 62 (2013), 271-283.  doi: 10.1080/02331934.2011.607497.  Google Scholar

[2]

G. BigiM. CastellaniM. Pappalardo and M. Passacantando, Existence and solution methods for equilibria, Eur. J. Oper. Res., 227 (2013), 1-11.  doi: 10.1016/j.ejor.2012.11.037.  Google Scholar

[3]

E. Blum and W. Oettli, From optimization and variational inequalities to equilibrium problems, Math. Student, 63 (1994), 127-149.   Google Scholar

[4]

P. L. Combettes and A. Hirstoaga, Equilibrium programming in Hilbert spaces, J. Nonlinear Convex Anal., 6 (2005), 117-136.   Google Scholar

[5]

J. ContrerasM. Klusch and J. B. Krawczyk, Numerical solution to Nash-Cournot equilibria in coupled constraint electricity markets, IEEE Trans. Power Syst., 19 (2004), 195-206.   Google Scholar

[6]

B. V. Dinh and D. S. Kim, Extragradient algorithms for equilibrium problems and symmetric generalized hybrid mappings, Optim. Lett., 11 (2016), 537-553.  doi: 10.1007/s11590-016-1025-5.  Google Scholar

[7]

B. V. Dinh and L. D. Muu, A projection algorithm for solving pseudomonotone equilibrium problems and it's application to a class of bilevel equilibria, Optim., 64 (2015), 559-575.  doi: 10.1080/02331934.2013.773329.  Google Scholar

[8]

B. V. DinhP. G. Hung and L. D. Muu, Bilevel optimization as a regularization approach to pseudomonotone equilibrium problems, Numer. Funct. Anal. Optim., 35 (2014), 539-563.  doi: 10.1080/01630563.2013.813857.  Google Scholar

[9]

B. V. Dinh and L. D. Muu, On penalty and gap function methods for bilevel equilibrium problems, J. Appl. Math., (2011) DOI: 10.1155/2011/646452. doi: 10.1155/2011/646452.  Google Scholar

[10]

F. Facchinei and J. S. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems, Springer, New York, 2003.  Google Scholar

[11]

A. Genel and J. Lindenstrass, An example concerning fixed points, Isarel J. Math., 22 (1975), 81-86.  doi: 10.1007/BF02757276.  Google Scholar

[12]

D. V. HieuL. D. Muu and P. K. Anh, Parallel hybrid extragradient methods for pseudomonotone equilibrium problems and nonexpansive mappings, Numer. Algor., 73 (2016), 197-217.  doi: 10.1007/s11075-015-0092-5.  Google Scholar

[13]

M. HojoT. Suzuki and W. Takahashi, Fixed point theorems and convergence theorems for generalized hybrid non-self mappings in Hilbert spaces, J. Nonlinear Convex Anal., 14 (2013), 363-376.   Google Scholar

[14]

H. Iiduka, Fixed point optimization algorithm and its application to power control in CDMA data networks, Math. Program., Ser. A, 133 (2012), 227-242.  doi: 10.1007/s10107-010-0427-x.  Google Scholar

[15]

S. Ishikawa, Fixed points by a new iteration method, Proc. Amer. Math. Soc., 40 (1974), 147-150.  doi: 10.2307/2039245.  Google Scholar

[16]

S. Itoh and W. Takahashi, The common fixed point theory of single-valued mappings and multi-valued mappings, Pacific J. Math., 79 (1978), 493-508.   Google Scholar

[17]

T. Kawasaki and W. Takahashi, Existence and mean approximation of fixed points of generalized hybrid mappings in Hilbert spaces, J. Nonlinear Convex Anal., 14 (2013), 71-87.   Google Scholar

[18]

F. Kohsaka and W. Takahashi, Fixed point theorems for a class of nonlinear mappings related to maximal monotone operators in Banach spaces, Arch. Math., 91 (2008), 166-177.  doi: 10.1007/s00013-008-2545-8.  Google Scholar

[19]

I. V. Konnov, Combined Relaxation Methods for Variational Inequalities, Lecture Notes in Economics and Mathematical System, 495 (2001), Springer, Berlin. doi: 10.1007/978-3-642-56886-2.  Google Scholar

[20]

G. M. Korpelevich, The extragradient method for finding saddle points and other problems, Matekon, 12 (1976), 747-756.   Google Scholar

[21]

W. R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc., 4 (1953), 506-510.  doi: 10.2307/2032162.  Google Scholar

[22]

F. Moradlou and Al izadeh, Strong convergence theorem by a new iterative method for equilibrium problems and symmetric generalized hybrid mappings, Mediterr. J. Math., 13 (2016), 379-390.  doi: 10.1007/s00009-014-0462-6.  Google Scholar

[23]

A. Moudafi, Proximal point algorithm extended to equilibrium problems, J. Nat. Geom., 15 (1999), 91-100.   Google Scholar

[24]

L. D. Muu and W. Oettli, Convergence of an adaptive penalty scheme for finding constrained equilibria, Nonlinear Anal.: TMA., 18 (1992), 1159-1166.  doi: 10.1016/0362-546X(92)90159-C.  Google Scholar

[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.  Google Scholar

[26] R. T. Rockafellar, Convex Analysis, Princeton University Press, 1970.   Google Scholar
[27]

J. Schu, Weak and strong convergence to fixed points of asymptotically noexpansive mappings, Bulletin of the Australian Math. Soc., 43 (1991), 153-159.  doi: 10.1017/S0004972700028884.  Google Scholar

[28]

A. Tada and W. Takahashi, Weak and strong convergence theorem for nonexpansive mapping and equilibrium problem, J. Optim. Theory Appl., 133 (2007), 359-370.  doi: 10.1007/s10957-007-9187-z.  Google Scholar

[29]

W. Takahashi and M. Toyoda, Weak convergence theorems for nonexpansive mappings and monotone mappings, J. Optim. Theory Appl., 118 (2003), 417-428.  doi: 10.1023/A:1025407607560.  Google Scholar

[30]

W. TakahashiN. C. Wong and J. C. Yao, Fixed point theorems for new generalized hybrid mappings in Hilbert spaces and applications, Taiwanese J. Math., 17 (2013), 1597-1611.  doi: 10.11650/tjm.17.2013.2921.  Google Scholar

[31]

W. Takahashi, Fixed point theorems for new nonlinear mappings in a Hilbert space, J. Nonlinear Convex Anal., 11 (2010), 79-88.   Google Scholar

[32]

D. Q. TranL. M. Dung and V. H. Nguyen, Extragradient algorithms extended to equilibrium problems, Optim., 57 (2008), 749-776.  doi: 10.1080/02331930601122876.  Google Scholar

[33]

N. N. TamJ. C. Yao and N. D. Yen, Solution methods for pseudomonotone variational inequalities, J. Optim. Theory Appl., 38 (2008), 253-273.  doi: 10.1007/s10957-008-9376-4.  Google Scholar

[34]

P. T. VuongJ. J. Strodiot and V. H. Nguyen, Extragradient methods and linesearch algorithms for solving Ky Fan inequalities and fixed point problems, J. Optim. Theory Appl., 155 (2013), 605-627.  doi: 10.1007/s10957-012-0085-7.  Google Scholar

[35]

H. K. Xu, A variable Krasnosel'ski$\mathop {\rm{i}}\limits^ \vee$-Mann algorithm and the multiple-set split feasibility problem, Inverse Problems, 22 (2006), 2021-2034.  doi: 10.1088/0266-5611/22/6/007.  Google Scholar

[36]

C. M. Yanes and H. K. Xu, Strong convergence of the $C Q$ method for fixed point iteration processes, Nonlinear Anal. TMA., 64 (2006), 2400-2411.  doi: 10.1016/j.na.2005.08.018.  Google Scholar

[37]

E. Zeidler, Nonlinear Functional Analysis and Its Applications I, Springer-Verlag, New York, 1986. doi: 10.1007/978-1-4612-4838-5.  Google Scholar

Table 1.  Results computed with Algorithm 2
N.P Size (n) Average Times Average Iterations
10 5 2.3766 152
10 10 4.2141 223
10 20 6.7813 457
10 30 10.5266 515
10 50 17.4891 567
10 100 29.2406 674
N.P Size (n) Average Times Average Iterations
10 5 2.3766 152
10 10 4.2141 223
10 20 6.7813 457
10 30 10.5266 515
10 50 17.4891 567
10 100 29.2406 674
Table 2.  Results computed with Algorithm 3
N.P Size (n) Average Times Average Iterations
10 5 2.4656 99
10 10 4.1422 132
10 20 6.6375 164
10 30 8.0672 170
10 50 11.8828 192
10 100 21.4953 210
N.P Size (n) Average Times Average Iterations
10 5 2.4656 99
10 10 4.1422 132
10 20 6.6375 164
10 30 8.0672 170
10 50 11.8828 192
10 100 21.4953 210
Table 3.  Results computed with Algorithm 1 in [6]
N.P Size (n) Average Times Average Iterations
10 5 23.2484 826
10 10 34.7438 1445
10 20 87.1016 2346
10 30 157.5781 2715
10 50 255.4578 3839
N.P Size (n) Average Times Average Iterations
10 5 23.2484 826
10 10 34.7438 1445
10 20 87.1016 2346
10 30 157.5781 2715
10 50 255.4578 3839
Table 4.  Results computed with Algorithm 2 in [6]
N.P Size (n) Average Times Average Iterations
10 5 38.5938 904
10 10 106.3172 2242
10 20 163.1266 3050
10 30 250.9313 3001
10 50 359.1094 3592
N.P Size (n) Average Times Average Iterations
10 5 38.5938 904
10 10 106.3172 2242
10 20 163.1266 3050
10 30 250.9313 3001
10 50 359.1094 3592
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