New algorithms for solving pseudo-monotone variational inequalities in Banach spaces

G. Reza Zamani Eskandanı; M. Raeisi; R. Lotfikar

doi:10.15672/hujms.1228124

Research Article

New algorithms for solving pseudo-monotone variational inequalities in Banach spaces

Year 2024, Volume: 53 Issue: 4, 981 - 1000, 27.08.2024

G. Reza Zamani Eskandanı , M. Raeisi , R. Lotfikar

https://doi.org/10.15672/hujms.1228124

Abstract

In this paper, we introduce new algorithms for finding a solution of a variational inequality problem involving pseudo-monotone operator which is also a fixed point of a Bregman relatively inexpensive mapping in $p$-uniformly convex and uniformly smooth Banach spaces that are more general than Hilbert spaces. We prove weak and strong convergence theorems for proposed algorithms. Finally, we give some numerical experiments for supporting our main results.

Keywords

variational inequality, pseudo-monotone operator, Bregman distance, Bregman projection

References

[1] Y. I. Alber, Metric and generalized projection operators in Banach spaces: Properties and applications, in: Theory and Applications of Nonlinear Operators of Accretive and Monotone Type Vol 178 of Lecture Notes in Pure and Applied Mathematics, New York: Dekker, 15-50, 1996.
[2] C. Baiocchi and A. Capelo, Variational and quasivariational inequalities: Applications to Free-Boundary Problems, John Wiley, New York, 1984.
[3] H. H. Bauschke and P. L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces, Springer, New York, 2011.
[4] D. Butnariu and E. Resmerita, Bregman distances, totally convex functions and a method for solving operator equations in Banach spaces, Abstr. Appl. Anal. 2006, Art. ID 084919, 1-39, 2006.
[5] D. Butnariu, I. N. Iusem and E. Resmerita, Totall convexity for powers of the norm in uniformly convex Banach spaces, J. Convex Anal. 7, 319-334, 2000.
[6] L. C. Ceng and J. C. Yao, Strong convergence theorem by an extragradient method for fixed point problems and variational inequality problems, Taiwan. J. Math. 10, 1293-1303, 2006.
[7] L. C. Ceng, N. Hadjisavvas and N. C. Wong, Strong convergence theorem by a hybrid extragradientlike approximation method for variational inequalities and fixed point problems, J. Glob. Optim. 46, 635-646, 2010.
[8] Y. Censor, A. Gibali and S. Reich, The subgradient extragradient method for solving variational inequalities in Hilbert space, J. Optim. Theory Appl. 148, 318-335, 2011.
[9] Y. Censor, A. Gibali and S. Reich, Extensions of Korpelevich’s extragradient method for the variational inequality problem in Euclidean space, Optimization 61, 1119-1132, 2011.
[10] Y. Censor, A. Gibali and S. Reich, Algorithms for the split variational inequality problem, Numer. Algorithms 56, 301-323, 2012.
[11] Y. Censor and A. Lent, An iterative row-action method for interval convex programming, J. Optim. Theory Appl. 34, 321-353, 1981.
[12] I. Cioranescu, Geometry of Banach spaces, duality mappings and nonlinear problems, Kluwer Academic Dordrecht, 1990.
[13] R. W. Cottle, J. C. Yao, Pseudo-monotone complementarity problems in Hilbert space, J. Optim. Theory Appl. 75, 281-295, 1992.
[14] G. Z. Eskandani, M. Raeisi and Th. M. Rassias,A hybrid extragradient method for solving pseudomonotone equilibrium problems using Bregman distance, J. Fixed Point Theory Appl. 20, 132, 2018.
[15] F. Facchinei and J. S. Pang, Finite-Dimensional variational inequalities and complementarity problems, Springer Series in Operations Research, vols. I and II, Springer, New York, 2003.
[16] C. J. Fang and Y. R. He, An extragradient method for generalized variational inequality, Pac. J. Optim. 9, 47-59, 2013.
[17] G. Fichera, Sul problema elastostatico di Signorini con ambigue condizioni al contorno, Rend. Accad. Naz. Lincei, s. VIII, 34, 1963.
[18] B. Halpern, Fixed points of nonexpanding maps. Bull. Am. Math. Soc. 73, 957-961, 1967.
[19] Y. R. He, A new double projection algorithm for variational inequalities, J. Comput. Appl. Math. 185, 166-173, 2006.
[20] X. Hu and J. Wang, Solving pseudo-monotone variational inequalities and pseudoconvex optimization problems using the projection neural network, IEEE Trans. Neural Netw. 17, 1487-1499, 2006.
[21] H. Iiduka and I. Yamada, A use of conjugate gradient direction for the convex optimization problem over the fixed point set of a nonexpansive mapping, SIAM J. Optim. 19, 1881-1893, 2008.
[22] H. Iiduka and I. Yamada, A subgradient-type method for the equilibrium problem over the fixed point set and its applications, Optimization 58, 251-261, 2009.
[23] A. N. Iusem, An iterative algorithm for the variational inequality problem, Comput. Appl. Math. 13, 103-114, 1994.
[24] A. N. Iusem and O. R. G´arciga, Inexact versions of proximal point and augmented Lagrangian algorithms in Banach spaces, Numer. Funct. Anal. Optim. 22, 609-640, 2001.
[25] A. N. Iusem and M. Nasri, Korpelevich’s method for variational inequality problems in Banach spaces, J. Global Optim. 50, 59-76, 2011.
[26] A. N. Iusem and B. F. Svaiter, A variant of Korpelevich’s method for variational inequalities with a new search strategy, Optimization 42(4), 309-321, 1997.
[27] S. Karamardian, Complementarity problems over cones with monotone and pseudomonotone maps, J. Optim. Theory Appl. 18, 445-454, 1976.
[28] D. Kinderlehrer and G. Stampacchia, An introduction to variational inequalities and their applications, Academic Press, New York, 1980.
[29] F. Kohsaka and W. Takahashi, Proximal point algorithm with Bregman functions in Banach spaces, J. Nonlinear Convex Anal. 6, 505-523, 2005.
[30] I. V. Konnov, Combined relaxation methods for variational inequalities, Springer Verlag, Berlin, Germany, 2001.
[31] G. M. Korpelevich, The extragradient method for finding saddle points and other problems, Ekon. Mat. Metody 12, 747-756, 1976.
[32] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces II, Springer, Berlin, 1979.
[33] P. E. Mainge, Projected subgradient techniques and viscosity methods for optimization with variational inequality constraints, Eur. J. Oper. Res. 205, 501-506, 2010.
[34] P. E. Maing´e, Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization, Set-valued Anal. 16, 899-912, 2008.
[35] Y.V. Malitsky and V. V. Semenov, A hybrid method without extrapolation step for solving variational inequality problems, J. Glob. Optim. 61, 193-202, 2015.
[36] N. Nadezhkina and W. Takahashi, Strong convergence theorem by a hybrid method for nonexpansive mappings and Lipschitz-continuous monotone mappings, SIAM J. Optim. 16, 1230-1241, 2006.
[37] N. Nadezhkina and W. Takahashi, Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings, J. Optim. Theory Appl. 128, 191-201, 2006.
[38] E. Naraghirad and J. C. Yao, Bregman weak relatively nonexpansive mappings in Banach spaces, Fixed Point Theory Appl. 2013, 141, 2013.
[39] Z. Opial, Weak convergence of the successive approximations for non expansive mappings in Banach spaces, Bull. Amer. Math. Soc. 73, 591-597, 1967.
[40] D. Reem, S. Reich and A. De Pierro, Re-examination of Bregman functions and new properties of their divergences, Optimization 68, 279-348, 2019.
[41] S. Reich, Book Review: Geometry of Banach spaces, duality mappings and nonlinear problems, Bull. Amer. Math. Soc. 26, 367-370, 1992.
[42] S. Reich, A weak convergence theorem for the alternating method with Bregman distances, in “Theory and Applications of Nonlinear Operators” 178, Marcel Dekker, New York, 313-318, 1996.
[43] S. Reich, D. V. Thong, L. Q. Dong, X. H. Li and V. T. Dung, New algorithms and convergence theorems for solving variational inequalities with non-Lipschitz mappings, Numerical Algorithms 87, 527-549, 2021.
[44] F. Schöpfer, T. Schuster and A. K. Louis, An iterative regularization method for the solution of the split feasibility problem in Banach spaces, Inverse Problems 24, 055008, 2008.
[45] G. Stampacchia, Variational inequalities, Theory and applications of monotone operators, Proceedings of the NATO Advanced Study Institute, Venice, Italy, 1968 (Edizioni Oderisi, Gubbio, Italy, 1968).
[46] M. Sibony, Methodes iteratives pour les equation set enequalitions aux derivees partielles nonlinearesde type monotone, Calcolo 7, 65-183, 1970.
[47] M. V. Solodov and B. F. Svaiter, A new projection method for variational inequality problems, SIAM J. Control Optim. 37(3), 765-776, 1999.
[48] W. Takahashi and M. Toyoda, Weak convergence theorems for nonexpansive mappings and monotone mappings, J. Optim. Theory Appl. 118, 417-428, 2003.
[49] D. V. Thong and D. V. Hieu, Modified subgradient extragradient algorithms for variational inequality problems and fixed point problems, Optimization 67, 83-102, 2018.
[50] D. V. Thong, Y. Shehu and O. S. Iyiola, Weak and strong convergence theorems for solving pseudo-monotone variational inequalities with non-Lipschitz mappings, Numerical Algorithms 84, 795-823, 2020.
[51] P. T. Vuong and Y. Shehu, Convergence of an extragradient-type method for variational inequality with applications to optimal control problems, Numer. Algorithms 81, 269-291, 2019.
[52] H. K. Xu, Another control condition in an iterative method for nonexpansive mappings, Bull. Austral. Math. Soc. 65, 109-113, 2002.
[53] M. L. Ye and Y. R. He, A double projection method for solving variational inequalities without mononicity, Comput. Optim. Appl. 60(1), 141-150, 2015.
[54] J. Zhao, Solving split equality fixed-point problem of quasi-nonexpansive mappings without prior knowledge of operators norms, Optimization 64, 2619-2630, 2015.

Year 2024, Volume: 53 Issue: 4, 981 - 1000, 27.08.2024

G. Reza Zamani Eskandanı , M. Raeisi , R. Lotfikar

https://doi.org/10.15672/hujms.1228124

Abstract

Keywords

Variational inequality, Pseudo-monotone operator, Bregman distance, Bregman projection

References

[1] Y. I. Alber, Metric and generalized projection operators in Banach spaces: Properties and applications, in: Theory and Applications of Nonlinear Operators of Accretive and Monotone Type Vol 178 of Lecture Notes in Pure and Applied Mathematics, New York: Dekker, 15-50, 1996.
[2] C. Baiocchi and A. Capelo, Variational and quasivariational inequalities: Applications to Free-Boundary Problems, John Wiley, New York, 1984.
[3] H. H. Bauschke and P. L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces, Springer, New York, 2011.
[4] D. Butnariu and E. Resmerita, Bregman distances, totally convex functions and a method for solving operator equations in Banach spaces, Abstr. Appl. Anal. 2006, Art. ID 084919, 1-39, 2006.
[5] D. Butnariu, I. N. Iusem and E. Resmerita, Totall convexity for powers of the norm in uniformly convex Banach spaces, J. Convex Anal. 7, 319-334, 2000.
[6] L. C. Ceng and J. C. Yao, Strong convergence theorem by an extragradient method for fixed point problems and variational inequality problems, Taiwan. J. Math. 10, 1293-1303, 2006.
[7] L. C. Ceng, N. Hadjisavvas and N. C. Wong, Strong convergence theorem by a hybrid extragradientlike approximation method for variational inequalities and fixed point problems, J. Glob. Optim. 46, 635-646, 2010.
[8] Y. Censor, A. Gibali and S. Reich, The subgradient extragradient method for solving variational inequalities in Hilbert space, J. Optim. Theory Appl. 148, 318-335, 2011.
[9] Y. Censor, A. Gibali and S. Reich, Extensions of Korpelevich’s extragradient method for the variational inequality problem in Euclidean space, Optimization 61, 1119-1132, 2011.
[10] Y. Censor, A. Gibali and S. Reich, Algorithms for the split variational inequality problem, Numer. Algorithms 56, 301-323, 2012.
[11] Y. Censor and A. Lent, An iterative row-action method for interval convex programming, J. Optim. Theory Appl. 34, 321-353, 1981.
[12] I. Cioranescu, Geometry of Banach spaces, duality mappings and nonlinear problems, Kluwer Academic Dordrecht, 1990.
[13] R. W. Cottle, J. C. Yao, Pseudo-monotone complementarity problems in Hilbert space, J. Optim. Theory Appl. 75, 281-295, 1992.
[14] G. Z. Eskandani, M. Raeisi and Th. M. Rassias,A hybrid extragradient method for solving pseudomonotone equilibrium problems using Bregman distance, J. Fixed Point Theory Appl. 20, 132, 2018.
[15] F. Facchinei and J. S. Pang, Finite-Dimensional variational inequalities and complementarity problems, Springer Series in Operations Research, vols. I and II, Springer, New York, 2003.
[16] C. J. Fang and Y. R. He, An extragradient method for generalized variational inequality, Pac. J. Optim. 9, 47-59, 2013.
[17] G. Fichera, Sul problema elastostatico di Signorini con ambigue condizioni al contorno, Rend. Accad. Naz. Lincei, s. VIII, 34, 1963.
[18] B. Halpern, Fixed points of nonexpanding maps. Bull. Am. Math. Soc. 73, 957-961, 1967.
[19] Y. R. He, A new double projection algorithm for variational inequalities, J. Comput. Appl. Math. 185, 166-173, 2006.
[20] X. Hu and J. Wang, Solving pseudo-monotone variational inequalities and pseudoconvex optimization problems using the projection neural network, IEEE Trans. Neural Netw. 17, 1487-1499, 2006.
[21] H. Iiduka and I. Yamada, A use of conjugate gradient direction for the convex optimization problem over the fixed point set of a nonexpansive mapping, SIAM J. Optim. 19, 1881-1893, 2008.
[22] H. Iiduka and I. Yamada, A subgradient-type method for the equilibrium problem over the fixed point set and its applications, Optimization 58, 251-261, 2009.
[23] A. N. Iusem, An iterative algorithm for the variational inequality problem, Comput. Appl. Math. 13, 103-114, 1994.
[24] A. N. Iusem and O. R. G´arciga, Inexact versions of proximal point and augmented Lagrangian algorithms in Banach spaces, Numer. Funct. Anal. Optim. 22, 609-640, 2001.
[25] A. N. Iusem and M. Nasri, Korpelevich’s method for variational inequality problems in Banach spaces, J. Global Optim. 50, 59-76, 2011.
[26] A. N. Iusem and B. F. Svaiter, A variant of Korpelevich’s method for variational inequalities with a new search strategy, Optimization 42(4), 309-321, 1997.
[27] S. Karamardian, Complementarity problems over cones with monotone and pseudomonotone maps, J. Optim. Theory Appl. 18, 445-454, 1976.
[28] D. Kinderlehrer and G. Stampacchia, An introduction to variational inequalities and their applications, Academic Press, New York, 1980.
[29] F. Kohsaka and W. Takahashi, Proximal point algorithm with Bregman functions in Banach spaces, J. Nonlinear Convex Anal. 6, 505-523, 2005.
[30] I. V. Konnov, Combined relaxation methods for variational inequalities, Springer Verlag, Berlin, Germany, 2001.
[31] G. M. Korpelevich, The extragradient method for finding saddle points and other problems, Ekon. Mat. Metody 12, 747-756, 1976.
[32] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces II, Springer, Berlin, 1979.
[33] P. E. Mainge, Projected subgradient techniques and viscosity methods for optimization with variational inequality constraints, Eur. J. Oper. Res. 205, 501-506, 2010.
[34] P. E. Maing´e, Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization, Set-valued Anal. 16, 899-912, 2008.
[35] Y.V. Malitsky and V. V. Semenov, A hybrid method without extrapolation step for solving variational inequality problems, J. Glob. Optim. 61, 193-202, 2015.
[36] N. Nadezhkina and W. Takahashi, Strong convergence theorem by a hybrid method for nonexpansive mappings and Lipschitz-continuous monotone mappings, SIAM J. Optim. 16, 1230-1241, 2006.
[37] N. Nadezhkina and W. Takahashi, Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings, J. Optim. Theory Appl. 128, 191-201, 2006.
[38] E. Naraghirad and J. C. Yao, Bregman weak relatively nonexpansive mappings in Banach spaces, Fixed Point Theory Appl. 2013, 141, 2013.
[39] Z. Opial, Weak convergence of the successive approximations for non expansive mappings in Banach spaces, Bull. Amer. Math. Soc. 73, 591-597, 1967.
[40] D. Reem, S. Reich and A. De Pierro, Re-examination of Bregman functions and new properties of their divergences, Optimization 68, 279-348, 2019.
[41] S. Reich, Book Review: Geometry of Banach spaces, duality mappings and nonlinear problems, Bull. Amer. Math. Soc. 26, 367-370, 1992.
[42] S. Reich, A weak convergence theorem for the alternating method with Bregman distances, in “Theory and Applications of Nonlinear Operators” 178, Marcel Dekker, New York, 313-318, 1996.
[43] S. Reich, D. V. Thong, L. Q. Dong, X. H. Li and V. T. Dung, New algorithms and convergence theorems for solving variational inequalities with non-Lipschitz mappings, Numerical Algorithms 87, 527-549, 2021.
[44] F. Schöpfer, T. Schuster and A. K. Louis, An iterative regularization method for the solution of the split feasibility problem in Banach spaces, Inverse Problems 24, 055008, 2008.
[45] G. Stampacchia, Variational inequalities, Theory and applications of monotone operators, Proceedings of the NATO Advanced Study Institute, Venice, Italy, 1968 (Edizioni Oderisi, Gubbio, Italy, 1968).
[46] M. Sibony, Methodes iteratives pour les equation set enequalitions aux derivees partielles nonlinearesde type monotone, Calcolo 7, 65-183, 1970.
[47] M. V. Solodov and B. F. Svaiter, A new projection method for variational inequality problems, SIAM J. Control Optim. 37(3), 765-776, 1999.
[48] W. Takahashi and M. Toyoda, Weak convergence theorems for nonexpansive mappings and monotone mappings, J. Optim. Theory Appl. 118, 417-428, 2003.
[49] D. V. Thong and D. V. Hieu, Modified subgradient extragradient algorithms for variational inequality problems and fixed point problems, Optimization 67, 83-102, 2018.
[50] D. V. Thong, Y. Shehu and O. S. Iyiola, Weak and strong convergence theorems for solving pseudo-monotone variational inequalities with non-Lipschitz mappings, Numerical Algorithms 84, 795-823, 2020.
[51] P. T. Vuong and Y. Shehu, Convergence of an extragradient-type method for variational inequality with applications to optimal control problems, Numer. Algorithms 81, 269-291, 2019.
[52] H. K. Xu, Another control condition in an iterative method for nonexpansive mappings, Bull. Austral. Math. Soc. 65, 109-113, 2002.
[53] M. L. Ye and Y. R. He, A double projection method for solving variational inequalities without mononicity, Comput. Optim. Appl. 60(1), 141-150, 2015.
[54] J. Zhao, Solving split equality fixed-point problem of quasi-nonexpansive mappings without prior knowledge of operators norms, Optimization 64, 2619-2630, 2015.

There are 54 citations in total.

Details

Primary Language	English
Subjects	Mathematical Sciences
Journal Section	Mathematics
Authors	G. Reza Zamani Eskandanı 0000-0001-8785-4778 M. Raeisi 0000-0001-9241-7353 R. Lotfikar 0000-0003-4550-9102
Early Pub Date	January 10, 2024
Publication Date	August 27, 2024
Published in Issue	Year 2024 Volume: 53 Issue: 4

Cite

APA	Zamani Eskandanı, G. R., Raeisi, M., & Lotfikar, R. (2024). New algorithms for solving pseudo-monotone variational inequalities in Banach spaces. Hacettepe Journal of Mathematics and Statistics, 53(4), 981-1000. https://doi.org/10.15672/hujms.1228124
AMA	Zamani Eskandanı GR, Raeisi M, Lotfikar R. New algorithms for solving pseudo-monotone variational inequalities in Banach spaces. Hacettepe Journal of Mathematics and Statistics. August 2024;53(4):981-1000. doi:10.15672/hujms.1228124
Chicago	Zamani Eskandanı, G. Reza, M. Raeisi, and R. Lotfikar. “New Algorithms for Solving Pseudo-Monotone Variational Inequalities in Banach Spaces”. Hacettepe Journal of Mathematics and Statistics 53, no. 4 (August 2024): 981-1000. https://doi.org/10.15672/hujms.1228124.
EndNote	Zamani Eskandanı GR, Raeisi M, Lotfikar R (August 1, 2024) New algorithms for solving pseudo-monotone variational inequalities in Banach spaces. Hacettepe Journal of Mathematics and Statistics 53 4 981–1000.
IEEE	G. R. Zamani Eskandanı, M. Raeisi, and R. Lotfikar, “New algorithms for solving pseudo-monotone variational inequalities in Banach spaces”, Hacettepe Journal of Mathematics and Statistics, vol. 53, no. 4, pp. 981–1000, 2024, doi: 10.15672/hujms.1228124.
ISNAD	Zamani Eskandanı, G. Reza et al. “New Algorithms for Solving Pseudo-Monotone Variational Inequalities in Banach Spaces”. Hacettepe Journal of Mathematics and Statistics 53/4 (August 2024), 981-1000. https://doi.org/10.15672/hujms.1228124.
JAMA	Zamani Eskandanı GR, Raeisi M, Lotfikar R. New algorithms for solving pseudo-monotone variational inequalities in Banach spaces. Hacettepe Journal of Mathematics and Statistics. 2024;53:981–1000.
MLA	Zamani Eskandanı, G. Reza et al. “New Algorithms for Solving Pseudo-Monotone Variational Inequalities in Banach Spaces”. Hacettepe Journal of Mathematics and Statistics, vol. 53, no. 4, 2024, pp. 981-1000, doi:10.15672/hujms.1228124.
Vancouver	Zamani Eskandanı GR, Raeisi M, Lotfikar R. New algorithms for solving pseudo-monotone variational inequalities in Banach spaces. Hacettepe Journal of Mathematics and Statistics. 2024;53(4):981-1000.

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