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Optimization problems with orthogonal matrix constraints
On the cyclic pseudomonotonicity and the proximal point algorithm
| 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 |
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 [
References:
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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. |
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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. |
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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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