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December  2018, 8(4): 441-449. doi: 10.3934/naco.2018027

On the cyclic pseudomonotonicity and the proximal point algorithm

Hadi Khatibzadeh 1,, , Vahid Mohebbi 1, and  Mohammad Hossein Alizadeh 2,

1. 

Department of Mathematics, University of Zanjan, P. O. Box 45195-313, Zanjan, Iran
2. 

Department of Mathematics, Institute for Advanced Studies in Basic Sciences, P. O. Box 45195-1159, Zanjan, Iran

* Corresponding author

Received  June 2017 Revised  August 2018 Published  September 2018

We introduce various versions of cyclic pseudomonotonicity and study the relations between them. Some examples about the relation between them and monotonicity are also presented. By imposing some assumptions on the cyclic pseudomonotone bifunctions, we study the convergence analysis of the proximal point algorithm which has been studied by Iusem and Sosa [5] for pseudomonotone bifunctions, with better assumptions.

Keywords: Equilibrium problem, n-pseudomonotone bifunction, cyclic pseudomonotone bifunction, proximal point algorithm, weak convergence, strong convergence.
Mathematics Subject Classification: 90C33, 65K15.
Citation: Hadi Khatibzadeh, Vahid Mohebbi, Mohammad Hossein Alizadeh. On the cyclic pseudomonotonicity and the proximal point algorithm. Numerical Algebra, Control and Optimization, 2018, 8 (4) : 441-449. doi: 10.3934/naco.2018027
References:
[1]

Z. Chbani and H. Riahi, Existence and asymptotic behaviour for solution of dynamical equilibrium systems, Evol. Equ. Control Theory, 3 (2014), 1-14.  doi: 10.3934/eect.2014.3.1.

[2]

N. Hadjisavvas and H. Khatibzadeh, Maximal monotonicity of bifunctions, Optimization, 59 (2010), 147-160.  doi: 10.1080/02331930801951116.

[3]

N. Hadjisavvas, S. Schaible and N. C. Wong, Pseudomonotone operator: a survey of the theory and its applications, J. Optim. Theory Appl., 152 (2012), 1-20.  doi: 10.1007/s10957-011-9912-5.

[4]

A. N. Iusem, G. Kassay and W. Sosa, On certain conditions for the existence of solutions of equilibrium problems, Math. Program., 116 (2009), 25-273.  doi: 10.1007/s10107-007-0125-5.

[5]

A. N. Iusem and W. Sosa, On the proximal point method for equilibrium problems in Hilbert spaces, Optimization, 59 (2010), 1259-1274.  doi: 10.1080/02331931003603133.

[6]

H. Khatibzadeh, V. Mohebbi and S. Ranjbar, Convergence analysis of the proximal point algorithm for pseudo-monotone equilibrium problems, Optim. Methods Softw., 30 (2015), 1146-1163.  doi: 10.1080/10556788.2015.1025402.

[7]

H. Khatibzadeh and V. Mohebbi, Proximal point algorithm for infinite pseudo-monotone bifunctions, Optimization, 65 (2016), 1629-1639.  doi: 10.1080/02331934.2016.1153639.

[8]

B. Martinet, Régularisation d'inéquations variationnelles par approximations successives, Rev. Française Informat. Recherche Opérationnelle, 3 (1970), 154-158. 

[9]

L. E. Muu, V. H. Nguyen and N. V. Quy, On Nash-Cournot oligopolistic market equilibrium models with concave cost functions, J. Global Optim., 41 (2008), 351-364.  doi: 10.1007/s10898-007-9243-0.

[10]

T. D. Quoc, P. N. Anh and L. D. Mu, Dual extragradient algorithms extended to equilibrium problems, J. Global Optim., 52 (2012), 139-159.  doi: 10.1007/s10898-011-9693-2.

[11]

R. T. Rockafellar, Monotone operators and the proximal point algorithm, SIAM J. Control Optim., 14 (1976), 877-898.  doi: 10.1137/0314056.

show all references

References:
[1]

Z. Chbani and H. Riahi, Existence and asymptotic behaviour for solution of dynamical equilibrium systems, Evol. Equ. Control Theory, 3 (2014), 1-14.  doi: 10.3934/eect.2014.3.1.

[2]

N. Hadjisavvas and H. Khatibzadeh, Maximal monotonicity of bifunctions, Optimization, 59 (2010), 147-160.  doi: 10.1080/02331930801951116.

[3]

N. Hadjisavvas, S. Schaible and N. C. Wong, Pseudomonotone operator: a survey of the theory and its applications, J. Optim. Theory Appl., 152 (2012), 1-20.  doi: 10.1007/s10957-011-9912-5.

[4]

A. N. Iusem, G. Kassay and W. Sosa, On certain conditions for the existence of solutions of equilibrium problems, Math. Program., 116 (2009), 25-273.  doi: 10.1007/s10107-007-0125-5.

[5]

A. N. Iusem and W. Sosa, On the proximal point method for equilibrium problems in Hilbert spaces, Optimization, 59 (2010), 1259-1274.  doi: 10.1080/02331931003603133.

[6]

H. Khatibzadeh, V. Mohebbi and S. Ranjbar, Convergence analysis of the proximal point algorithm for pseudo-monotone equilibrium problems, Optim. Methods Softw., 30 (2015), 1146-1163.  doi: 10.1080/10556788.2015.1025402.

[7]

H. Khatibzadeh and V. Mohebbi, Proximal point algorithm for infinite pseudo-monotone bifunctions, Optimization, 65 (2016), 1629-1639.  doi: 10.1080/02331934.2016.1153639.

[8]

B. Martinet, Régularisation d'inéquations variationnelles par approximations successives, Rev. Française Informat. Recherche Opérationnelle, 3 (1970), 154-158. 

[9]

L. E. Muu, V. H. Nguyen and N. V. Quy, On Nash-Cournot oligopolistic market equilibrium models with concave cost functions, J. Global Optim., 41 (2008), 351-364.  doi: 10.1007/s10898-007-9243-0.

[10]

T. D. Quoc, P. N. Anh and L. D. Mu, Dual extragradient algorithms extended to equilibrium problems, J. Global Optim., 52 (2012), 139-159.  doi: 10.1007/s10898-011-9693-2.

[11]

R. T. Rockafellar, Monotone operators and the proximal point algorithm, SIAM J. Control Optim., 14 (1976), 877-898.  doi: 10.1137/0314056.

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