Inertial Tseng's extragradient method for solving variational inequality problems of pseudo-monotone and non-Lipschitz operators

Gang Cai; Yekini Shehu; Olaniyi S. Iyiola

doi:10.3934/jimo.2021095

Previous Article
Optimal pricing, ordering, and credit period policies for deteriorating products under order-linked trade credit
JIMO Home
This Issue
Next Article
Using optimal control to optimize the extraction rate of a durable non-renewable resource with a monopolistic primary supplier

doi: 10.3934/jimo.2021095

Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

Inertial Tseng's extragradient method for solving variational inequality problems of pseudo-monotone and non-Lipschitz operators

Gang Cai ^1,, , Yekini Shehu ^2, and Olaniyi S. Iyiola ^3,

1.	School of Mathematics Science, Chongqing Normal University, Chongqing 401331, China
2.	Department of Mathematics, Zhejiang Normal University, Jinhua, 321004, China
3.	Department of Mathematics and Physical Sciences, , California University of Pennsylvania, PA, USA

^* Corresponding author: G. Cai

Received September 2020 Revised January 2021 Early access May 2021

Fund Project: The first author is supported by the NSF of China (Grant No. 11771063), the Natural Science Foundation of Chongqing(cstc2020jcyj-msxmX0455), Science and Technology Project of Chongqing Education Committee (Grant No. KJZD-K201900504) and the Program of Chongqing Innovation Research Group Project in University (Grant no. CXQT19018)

In this paper, we propose a new inertial Tseng's extragradient iterative algorithm for solving variational inequality problems of pseudo-monotone and non-Lipschitz operator in real Hilbert spaces. We prove that the sequence generated by proposed algorithm converges strongly to an element of solutions of variational inequality problem under some suitable assumptions imposed on the parameters. Finally, we give some numerical experiments for supporting our main results. The main results obtained in this paper extend and improve some related works in the literature.

Keywords: Tseng's extragradient method, variational inequality, pseudo-monotone operators, strong convergence.

Mathematics Subject Classification: Primary: 47H09, 47H10; Secondary: 47J20, 65K15.

Citation: Gang Cai, Yekini Shehu, Olaniyi S. Iyiola. Inertial Tseng's extragradient method for solving variational inequality problems of pseudo-monotone and non-Lipschitz operators. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2021095

References:

[1]	T. O. Alakoya, L. O. Jolaoso and O. T. Mewomo, Modified inertial subgradient extragradient method with self adaptive stepsize for solving monotone variational inequality and fixed point problems, Optimization, 70 (2021), 545-574. doi: 10.1080/02331934.2020.1723586.
[2]	F. Alvarez and H. Attouch, An inertial proximal method for maximal monotone operators via discretization of a nonlinear oscillator with damping, Set. Valued Anal., 9 (2001), 3-11. doi: 10.1023/A:1011253113155.
[3]	F. Alvarez, Weak convergence of a relaxed and inertial hybrid projection proximal point algorithm for maximal monotone operators in Hilbert space, SIAM J. Optim., 14 (2004), 773-782. doi: 10.1137/S1052623403427859.
[4]	C. Baiocchi and A. Capelo, Variational and quasivariational inequalities: Applications to free boundary problems, Wiley, New York, 1984.
[5]	R. I. Bǫt, E. R. Csetnek and A. Heinrich, A primal-dual splitting algorithm for finding zeros of sums of maximally monotone operators, SIAM J. Optim., 23 (2013), 2011-2036. doi: 10.1137/12088255X.
[6]	R. I. Bǫt and C. Hendrich, A Douglas-Rachford type primal-dual method for solving inclusions with mixtures of composite and parallel-sum type monotone operators, SIAM J. Optim., 23 (2013), 2541-2565. doi: 10.1137/120901106.
[7]	R. I. Bǫt, E. R. Csetnek, A. Heinrich and C. Hendrich, On the convergence rate improvement of a primal-dual splitting algorithm for solving monotone inclusion problems, Math. Program., 150 (2015), 251-279. doi: 10.1007/s10107-014-0766-0.
[8]	R. I. Bǫt, E. R. Csetnek and C. Hendrich, Inertial Douglas-Rachford splitting for monotone inclusion problems, Appl. Math. Comput., 256 (2015), 472-487. doi: 10.1016/j.amc.2015.01.017.
[9]	R. I. Bǫt and E. R. Csetnek, An inertial Tseng's type proximal algorithm for nonsmooth and nonconvex optimization problems, J. Optim. Theory Appl., 171 (2016), 600-616. doi: 10.1007/s10957-015-0730-z.
[10]	R. I. Bǫt, E. R. Csetnek and P. T. Vuong, The forward-backward-forward method from continuous and discrete perspective for pseudo-monotone variational inequalities in Hilbert spaces, European J. Oper. Res., 287 (2020), 49-60. doi: 10.1016/j.ejor.2020.04.035.
[11]	Y. Censor, A. Gibali and S. Reich, The subgradient extragradient method for solving variational inequalities in Hilbert space, J. Optim. Theory Appl., 148 (2011), 318-335. doi: 10.1007/s10957-010-9757-3.
[12]	Y. Censor, A. Gibali and S. Reich, Extensions of Korpelevich's extragradient method for the variational inequality problem in Euclidean space, Optimization, 61 (2011), 1119-1132. doi: 10.1080/02331934.2010.539689.
[13]	Y. Censor, A. Gibali and S. Reich, Algorithms for the split variational inequality problem, Numer. Algorithms, 56 (2012), 301-323. doi: 10.1007/s11075-011-9490-5.
[14]	R. W. Cottle and J. C. Yao, Pseudo-monotone complementarity problems in Hilbert space, J. Optim. Theory Appl., 75 (1992), 281-295. doi: 10.1007/BF00941468.
[15]	S. V. Denisov, V. V. Semenov and L. M. Chabak, Convergence of the modified extragradient method for variational inequalities with non-Lipschitz operators, Cybern. Syst. Anal., 51 (2015), 757-765. doi: 10.1007/s10559-015-9768-z.
[16]	Q. L. Dong, Y. J. Cho, L. L. Zhong and Th. M. Rassias, Inertial projection and contraction algorithms for variational inequalities, J. Glob. Optim., 70 (2018), 687-704. doi: 10.1007/s10898-017-0506-0.
[17]	Q. L. Dong, H. B.Yuan, Y. J. Cho and Th. M. Rassias, Modified inertial Mann algorithm and inertial CQ-algorithm for nonexpansive mappings, Optim. Lett., 12 (2018), 87-102. doi: 10.1007/s11590-016-1102-9.
[18]	Q.-L. Dong, K. R. Kazmi, R. Ali and X.-H. Li, Inertial Krasnosel'skii-Mann type hybrid algorithms for solving hierarchical fixed point problems, J. Fixed Point Theory Appl., 21 (2019), 57. doi: 10.1007/s11784-019-0699-6.
[19]	F. Facchinei and J. S. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems, Springer Series in Operations Research, vol. II. Springer, New York, 2003.
[20]	A. Gibali, D. V. Thong and P. A. Tuan, Two simple projection-type methods for solving variational inequalities, Anal. Math. Phys., 9 (2019), 2203-2225. doi: 10.1007/s13324-019-00330-w.
[21]	R. Glowinski, J.-L. Lions and R. Trémolières, Numerical Analysis of Variational Inequalities, Elsevier, Amsterdam, 1981.
[22]	K. Goebel and S. Reich, Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, Marcel Dekker, New York, 1984.
[23]	P. T. Harker and J.-S. Pang, A damped-newton method for the linear complementarity problem, Lect. Appl. Math., 26 (1990), 265-284. doi: 10.1007/bf01582255.
[24]	X. Hu and J. Wang, Solving pseudo-monotone variational inequalities and pseudo-convex optimization problems using the projection neural network, IEEE Trans. Neural Netw., 17 (2006), 1487-1499.
[25]	A. N. Iusem, An iterative algorithm for the variational inequality problem, Mat. Apl. Comput., 13 (1994), 103-114.
[26]	A. Iusem and R. Gárciga Otero, Inexact versions of proximal point and augmented Lagrangian algorithms in Banach spaces, Numer. Funct. Anal. Optim., 22 (2001), 609-640. doi: 10.1081/NFA-100105310.
[27]	C. Izuchukwu, A. A. Mebawondu and O. T. Mewomo, A new method for solving split variational inequality problem without co-coerciveness, J. Fixed Point Theory Appl., 22 (2020), 98. doi: 10.1007/s11784-020-00834-0.
[28]	L. O. Jolaoso, A. Taiwo, T. O. Alakoya and O. T. Mewomo, A strong convergence theorem for solving pseudo-monotone variational inequalities using projection methods, J. Optim. Theory Appl., 185 (2020), 744-766. doi: 10.1007/s10957-020-01672-3.
[29]	L. O. Jolaoso, A. Taiwo, T. O. Alakoya and O. T. Mewomo, A unified algorithm for solving variational inequality and fixed point problems with application to the split equality problem, Comput. Appl. Math., 39 (2020), 38. doi: 10.1007/s40314-019-1014-2.
[30]	L. O. Jolaoso, T. O. Alakoya, A. Taiwo and O. T. Mewomo, Inertial extragradient method via viscosity approximation approach for solving Equilibrium problem in Hilbert space, Optimization, 70 (2021), 387-412. doi: 10.1080/02331934.2020.1716752.
[31]	G. Kassay, S. Reich and S. Sabach, Iterative methods for solving systems of variational inequalities in refelexive Banach spaces, SIAM J. Optim., 21 (2011), 1319-1344. doi: 10.1137/110820002.
[32]	D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and their Applications, Academic Press, New York, 1980.
[33]	I. Konnov, Combined Relaxation Methods for Variational Inequalities, Springer, Berlin, 2001. doi: 10.1007/978-3-642-56886-2.
[34]	G. M. Korpelevič, The extragradient method for finding saddle points and other problems, Ekon. Mat. Metody, 12 (1976), 747-756.
[35]	P.-E. Maingé, A hybrid extragradient-viscosity method for monotone operators and fixed point problems, SIAM J. Control Optim., 47 (2008), 1499-1515. doi: 10.1137/060675319.
[36]	P.-E. Maingé, Convergence theorem for inertial KM-type algorithms, J. Comput. Appl. Math., 219 (2008), 223-236. doi: 10.1016/j.cam.2007.07.021.
[37]	Y. Malitsky, Projected reflected gradient methods for monotone variational inequalities, SIAM J. Optim., 25 (2015), 502-520. doi: 10.1137/14097238X.
[38]	Y. V. Malitsky and V. V. Semenov, A hybrid method without extrapolation step for solving variational inequality problems, J. Glob. Optim., 61 (2015), 193-202. doi: 10.1007/s10898-014-0150-x.
[39]	Y. Shehu and O. S. Iyiola, Convergence analysis for the proximal split feasibility problem using an inertial extrapolation term method, J. Fixed Point Theory Appl., 19 (2017), 2483-2510. doi: 10.1007/s11784-017-0435-z.
[40]	Y. Shehu, O. S. Iyiola and F. U. Ogbuisi, Iterative method with inertial terms for nonexpansive mappings: Applications to compressed sensing, Numer Algorithms, 83 (2020), 1321-1347. doi: 10.1007/s11075-019-00727-5.
[41]	Y. Shehu and P. Cholamjiak, Iterative method with inertial for variational inequalities in Hilbert spaces, Calcolo, 56 (2019), 4. doi: 10.1007/s10092-018-0300-5.
[42]	Y. Shuhu, Q.-L. Dong and D. Jiang, Single projection method for pseudo-monotone variational inequality in Hilbert spaces, Optimization, 68 (2019), 385-409. doi: 10.1080/02331934.2018.1522636.
[43]	M. V. Solodov and P. Tseng, Modified projection-type methods for monotone variational inequalities, SIAM J. Control Optim., 34 (1996), 1814-1830. doi: 10.1137/S0363012994268655.
[44]	M. V. Solodov and B. F. Svaiter, A new projection method for variational inequality problems, SIAM J. Control Optim., 37 (1999), 765-776. doi: 10.1137/S0363012997317475.
[45]	D. V. Thong and D. V. Hieu, Inertial extragradient algorithms for strongly pseudomonotone variational inequalities, J. Comput. Appl. Math., 341 (2018), 80-98. doi: 10.1016/j.cam.2018.03.019.
[46]	D. V. Thong and D. V.Hieu, Modified subgradient extragradient method for variational inequality problems, Numer. Algorithms, 79 (2018), 597-610. doi: 10.1007/s11075-017-0452-4.
[47]	D. V. Thong and D. V. Hieu, Weak and strong convergence theorems for variational inequality problems, Numer. Algorithms, 78 (2018), 1045-1060. doi: 10.1007/s11075-017-0412-z.
[48]	D. V. Thong and D. V. Hieu, Inertial subgradient extragradient algorithms with line-search process for solving variational inequality problems and fixed point problems, Numer Algorithms, 80 (2019), 1283-1307. doi: 10.1007/s11075-018-0527-x.
[49]	D. V. Thong, N. T. Vinh and Y. J. Cho, Accelerated subgradient extragradient methods for variational inequality problems, J. Sci. Comput., 80 (2019), 1438-1462. doi: 10.1007/s10915-019-00984-5.
[50]	D. V. Thong, D. Van Hieu and T. M. Rassias, Self adaptive inertial subgradient extragradient algorithms for solving pseudomonotone variational inequality problems, Optim. Lett., 14 (2020), 115-144. doi: 10.1007/s11590-019-01511-z.
[51]	D. V. Thong and A. Gibali, Extragradient methods for solving non-Lipschitzian pseudo-monotone variational inequalities, J. Fixed Point Theory Appl., 21 (2019), 20. doi: 10.1007/s11784-018-0656-9.
[52]	D. V. Thong, N. T. Vinh and Y. J. Cho, New strong convergence theorem of the inertial projection and contraction method for variational inequality problems, Numer Algorithms, 84 (2020), 285-305. doi: 10.1007/s11075-019-00755-1.
[53]	D. V. Thong and D. Van Hieu, Strong convergence of extragradient methods with a new step size for solving variational inequality problems, Comp. Appl. Math., 38 (2019), 136. doi: 10.1007/s40314-019-0899-0.
[54]	D. V. Thong, Y. Shehu and O. S. Iyiola, Weak and strong convergence theorems for solving pseudo-monotone variational inequalities with non-Lipschitz mappings, Numer. Algorithms, 84 (2020), 795-823. doi: 10.1007/s11075-019-00780-0.
[55]	D. V. Thong and D. V. Hieu, New extragradient methods for solving variational inequality problems and fixed point problems, J. Fixed Point Theory Appl., 20 (2018), 129. doi: 10.1007/s11784-018-0610-x.
[56]	D. V. Thong and D. V. Hieu, Modified Tseng's extragradient algorithms for variational inequality problems, J. Fixed Point Theory Appl., 20 (2018), 152. doi: 10.1007/s11784-018-0634-2.
[57]	D. V. Thong, N. A. Triet, X.-H. Li and Q.-L. Dong, Strong convergence of extragradient methods for solving bilevel pseudo-monotone variational inequality problems, Numer. Algorithms, 83 (2020), 1123-1143. doi: 10.1007/s11075-019-00718-6.
[58]	P. Tseng, A modified forward-backward splitting method for maximal monotone mappings, SIAM J. Control Optim., 38 (2000), 431-446. doi: 10.1137/S0363012998338806.
[59]	F. Wang and H.-K. Xu, Weak and strong convergence theorems for variational inequality and fixed point problems with Tseng's extragradient method, Taiwan. J. Math., 16 (2012), 1125-1136. doi: 10.11650/twjm/1500406682.
[60]	H.-K. Xu, Iterative algorithms for nonlinear operators, J. Lond. Math. Soc., 66 (2002), 240-256. doi: 10.1112/S0024610702003332.

show all references

References:

[1]	T. O. Alakoya, L. O. Jolaoso and O. T. Mewomo, Modified inertial subgradient extragradient method with self adaptive stepsize for solving monotone variational inequality and fixed point problems, Optimization, 70 (2021), 545-574. doi: 10.1080/02331934.2020.1723586.
[2]	F. Alvarez and H. Attouch, An inertial proximal method for maximal monotone operators via discretization of a nonlinear oscillator with damping, Set. Valued Anal., 9 (2001), 3-11. doi: 10.1023/A:1011253113155.
[3]	F. Alvarez, Weak convergence of a relaxed and inertial hybrid projection proximal point algorithm for maximal monotone operators in Hilbert space, SIAM J. Optim., 14 (2004), 773-782. doi: 10.1137/S1052623403427859.
[4]	C. Baiocchi and A. Capelo, Variational and quasivariational inequalities: Applications to free boundary problems, Wiley, New York, 1984.
[5]	R. I. Bǫt, E. R. Csetnek and A. Heinrich, A primal-dual splitting algorithm for finding zeros of sums of maximally monotone operators, SIAM J. Optim., 23 (2013), 2011-2036. doi: 10.1137/12088255X.
[6]	R. I. Bǫt and C. Hendrich, A Douglas-Rachford type primal-dual method for solving inclusions with mixtures of composite and parallel-sum type monotone operators, SIAM J. Optim., 23 (2013), 2541-2565. doi: 10.1137/120901106.
[7]	R. I. Bǫt, E. R. Csetnek, A. Heinrich and C. Hendrich, On the convergence rate improvement of a primal-dual splitting algorithm for solving monotone inclusion problems, Math. Program., 150 (2015), 251-279. doi: 10.1007/s10107-014-0766-0.
[8]	R. I. Bǫt, E. R. Csetnek and C. Hendrich, Inertial Douglas-Rachford splitting for monotone inclusion problems, Appl. Math. Comput., 256 (2015), 472-487. doi: 10.1016/j.amc.2015.01.017.
[9]	R. I. Bǫt and E. R. Csetnek, An inertial Tseng's type proximal algorithm for nonsmooth and nonconvex optimization problems, J. Optim. Theory Appl., 171 (2016), 600-616. doi: 10.1007/s10957-015-0730-z.
[10]	R. I. Bǫt, E. R. Csetnek and P. T. Vuong, The forward-backward-forward method from continuous and discrete perspective for pseudo-monotone variational inequalities in Hilbert spaces, European J. Oper. Res., 287 (2020), 49-60. doi: 10.1016/j.ejor.2020.04.035.
[11]	Y. Censor, A. Gibali and S. Reich, The subgradient extragradient method for solving variational inequalities in Hilbert space, J. Optim. Theory Appl., 148 (2011), 318-335. doi: 10.1007/s10957-010-9757-3.
[12]	Y. Censor, A. Gibali and S. Reich, Extensions of Korpelevich's extragradient method for the variational inequality problem in Euclidean space, Optimization, 61 (2011), 1119-1132. doi: 10.1080/02331934.2010.539689.
[13]	Y. Censor, A. Gibali and S. Reich, Algorithms for the split variational inequality problem, Numer. Algorithms, 56 (2012), 301-323. doi: 10.1007/s11075-011-9490-5.
[14]	R. W. Cottle and J. C. Yao, Pseudo-monotone complementarity problems in Hilbert space, J. Optim. Theory Appl., 75 (1992), 281-295. doi: 10.1007/BF00941468.
[15]	S. V. Denisov, V. V. Semenov and L. M. Chabak, Convergence of the modified extragradient method for variational inequalities with non-Lipschitz operators, Cybern. Syst. Anal., 51 (2015), 757-765. doi: 10.1007/s10559-015-9768-z.
[16]	Q. L. Dong, Y. J. Cho, L. L. Zhong and Th. M. Rassias, Inertial projection and contraction algorithms for variational inequalities, J. Glob. Optim., 70 (2018), 687-704. doi: 10.1007/s10898-017-0506-0.
[17]	Q. L. Dong, H. B.Yuan, Y. J. Cho and Th. M. Rassias, Modified inertial Mann algorithm and inertial CQ-algorithm for nonexpansive mappings, Optim. Lett., 12 (2018), 87-102. doi: 10.1007/s11590-016-1102-9.
[18]	Q.-L. Dong, K. R. Kazmi, R. Ali and X.-H. Li, Inertial Krasnosel'skii-Mann type hybrid algorithms for solving hierarchical fixed point problems, J. Fixed Point Theory Appl., 21 (2019), 57. doi: 10.1007/s11784-019-0699-6.
[19]	F. Facchinei and J. S. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems, Springer Series in Operations Research, vol. II. Springer, New York, 2003.
[20]	A. Gibali, D. V. Thong and P. A. Tuan, Two simple projection-type methods for solving variational inequalities, Anal. Math. Phys., 9 (2019), 2203-2225. doi: 10.1007/s13324-019-00330-w.
[21]	R. Glowinski, J.-L. Lions and R. Trémolières, Numerical Analysis of Variational Inequalities, Elsevier, Amsterdam, 1981.
[22]	K. Goebel and S. Reich, Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, Marcel Dekker, New York, 1984.
[23]	P. T. Harker and J.-S. Pang, A damped-newton method for the linear complementarity problem, Lect. Appl. Math., 26 (1990), 265-284. doi: 10.1007/bf01582255.
[24]	X. Hu and J. Wang, Solving pseudo-monotone variational inequalities and pseudo-convex optimization problems using the projection neural network, IEEE Trans. Neural Netw., 17 (2006), 1487-1499.
[25]	A. N. Iusem, An iterative algorithm for the variational inequality problem, Mat. Apl. Comput., 13 (1994), 103-114.
[26]	A. Iusem and R. Gárciga Otero, Inexact versions of proximal point and augmented Lagrangian algorithms in Banach spaces, Numer. Funct. Anal. Optim., 22 (2001), 609-640. doi: 10.1081/NFA-100105310.
[27]	C. Izuchukwu, A. A. Mebawondu and O. T. Mewomo, A new method for solving split variational inequality problem without co-coerciveness, J. Fixed Point Theory Appl., 22 (2020), 98. doi: 10.1007/s11784-020-00834-0.
[28]	L. O. Jolaoso, A. Taiwo, T. O. Alakoya and O. T. Mewomo, A strong convergence theorem for solving pseudo-monotone variational inequalities using projection methods, J. Optim. Theory Appl., 185 (2020), 744-766. doi: 10.1007/s10957-020-01672-3.
[29]	L. O. Jolaoso, A. Taiwo, T. O. Alakoya and O. T. Mewomo, A unified algorithm for solving variational inequality and fixed point problems with application to the split equality problem, Comput. Appl. Math., 39 (2020), 38. doi: 10.1007/s40314-019-1014-2.
[30]	L. O. Jolaoso, T. O. Alakoya, A. Taiwo and O. T. Mewomo, Inertial extragradient method via viscosity approximation approach for solving Equilibrium problem in Hilbert space, Optimization, 70 (2021), 387-412. doi: 10.1080/02331934.2020.1716752.
[31]	G. Kassay, S. Reich and S. Sabach, Iterative methods for solving systems of variational inequalities in refelexive Banach spaces, SIAM J. Optim., 21 (2011), 1319-1344. doi: 10.1137/110820002.
[32]	D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and their Applications, Academic Press, New York, 1980.
[33]	I. Konnov, Combined Relaxation Methods for Variational Inequalities, Springer, Berlin, 2001. doi: 10.1007/978-3-642-56886-2.
[34]	G. M. Korpelevič, The extragradient method for finding saddle points and other problems, Ekon. Mat. Metody, 12 (1976), 747-756.
[35]	P.-E. Maingé, A hybrid extragradient-viscosity method for monotone operators and fixed point problems, SIAM J. Control Optim., 47 (2008), 1499-1515. doi: 10.1137/060675319.
[36]	P.-E. Maingé, Convergence theorem for inertial KM-type algorithms, J. Comput. Appl. Math., 219 (2008), 223-236. doi: 10.1016/j.cam.2007.07.021.
[37]	Y. Malitsky, Projected reflected gradient methods for monotone variational inequalities, SIAM J. Optim., 25 (2015), 502-520. doi: 10.1137/14097238X.
[38]	Y. V. Malitsky and V. V. Semenov, A hybrid method without extrapolation step for solving variational inequality problems, J. Glob. Optim., 61 (2015), 193-202. doi: 10.1007/s10898-014-0150-x.
[39]	Y. Shehu and O. S. Iyiola, Convergence analysis for the proximal split feasibility problem using an inertial extrapolation term method, J. Fixed Point Theory Appl., 19 (2017), 2483-2510. doi: 10.1007/s11784-017-0435-z.
[40]	Y. Shehu, O. S. Iyiola and F. U. Ogbuisi, Iterative method with inertial terms for nonexpansive mappings: Applications to compressed sensing, Numer Algorithms, 83 (2020), 1321-1347. doi: 10.1007/s11075-019-00727-5.
[41]	Y. Shehu and P. Cholamjiak, Iterative method with inertial for variational inequalities in Hilbert spaces, Calcolo, 56 (2019), 4. doi: 10.1007/s10092-018-0300-5.
[42]	Y. Shuhu, Q.-L. Dong and D. Jiang, Single projection method for pseudo-monotone variational inequality in Hilbert spaces, Optimization, 68 (2019), 385-409. doi: 10.1080/02331934.2018.1522636.
[43]	M. V. Solodov and P. Tseng, Modified projection-type methods for monotone variational inequalities, SIAM J. Control Optim., 34 (1996), 1814-1830. doi: 10.1137/S0363012994268655.
[44]	M. V. Solodov and B. F. Svaiter, A new projection method for variational inequality problems, SIAM J. Control Optim., 37 (1999), 765-776. doi: 10.1137/S0363012997317475.
[45]	D. V. Thong and D. V. Hieu, Inertial extragradient algorithms for strongly pseudomonotone variational inequalities, J. Comput. Appl. Math., 341 (2018), 80-98. doi: 10.1016/j.cam.2018.03.019.
[46]	D. V. Thong and D. V.Hieu, Modified subgradient extragradient method for variational inequality problems, Numer. Algorithms, 79 (2018), 597-610. doi: 10.1007/s11075-017-0452-4.
[47]	D. V. Thong and D. V. Hieu, Weak and strong convergence theorems for variational inequality problems, Numer. Algorithms, 78 (2018), 1045-1060. doi: 10.1007/s11075-017-0412-z.
[48]	D. V. Thong and D. V. Hieu, Inertial subgradient extragradient algorithms with line-search process for solving variational inequality problems and fixed point problems, Numer Algorithms, 80 (2019), 1283-1307. doi: 10.1007/s11075-018-0527-x.
[49]	D. V. Thong, N. T. Vinh and Y. J. Cho, Accelerated subgradient extragradient methods for variational inequality problems, J. Sci. Comput., 80 (2019), 1438-1462. doi: 10.1007/s10915-019-00984-5.
[50]	D. V. Thong, D. Van Hieu and T. M. Rassias, Self adaptive inertial subgradient extragradient algorithms for solving pseudomonotone variational inequality problems, Optim. Lett., 14 (2020), 115-144. doi: 10.1007/s11590-019-01511-z.
[51]	D. V. Thong and A. Gibali, Extragradient methods for solving non-Lipschitzian pseudo-monotone variational inequalities, J. Fixed Point Theory Appl., 21 (2019), 20. doi: 10.1007/s11784-018-0656-9.
[52]	D. V. Thong, N. T. Vinh and Y. J. Cho, New strong convergence theorem of the inertial projection and contraction method for variational inequality problems, Numer Algorithms, 84 (2020), 285-305. doi: 10.1007/s11075-019-00755-1.
[53]	D. V. Thong and D. Van Hieu, Strong convergence of extragradient methods with a new step size for solving variational inequality problems, Comp. Appl. Math., 38 (2019), 136. doi: 10.1007/s40314-019-0899-0.
[54]	D. V. Thong, Y. Shehu and O. S. Iyiola, Weak and strong convergence theorems for solving pseudo-monotone variational inequalities with non-Lipschitz mappings, Numer. Algorithms, 84 (2020), 795-823. doi: 10.1007/s11075-019-00780-0.
[55]	D. V. Thong and D. V. Hieu, New extragradient methods for solving variational inequality problems and fixed point problems, J. Fixed Point Theory Appl., 20 (2018), 129. doi: 10.1007/s11784-018-0610-x.
[56]	D. V. Thong and D. V. Hieu, Modified Tseng's extragradient algorithms for variational inequality problems, J. Fixed Point Theory Appl., 20 (2018), 152. doi: 10.1007/s11784-018-0634-2.
[57]	D. V. Thong, N. A. Triet, X.-H. Li and Q.-L. Dong, Strong convergence of extragradient methods for solving bilevel pseudo-monotone variational inequality problems, Numer. Algorithms, 83 (2020), 1123-1143. doi: 10.1007/s11075-019-00718-6.
[58]	P. Tseng, A modified forward-backward splitting method for maximal monotone mappings, SIAM J. Control Optim., 38 (2000), 431-446. doi: 10.1137/S0363012998338806.
[59]	F. Wang and H.-K. Xu, Weak and strong convergence theorems for variational inequality and fixed point problems with Tseng's extragradient method, Taiwan. J. Math., 16 (2012), 1125-1136. doi: 10.11650/twjm/1500406682.
[60]	H.-K. Xu, Iterative algorithms for nonlinear operators, J. Lond. Math. Soc., 66 (2002), 240-256. doi: 10.1112/S0024610702003332.

Figure 1. Example 1: $ k = 20 $, $ N = 10 $

Proposed Alg.	$ \epsilon_n = \frac{1}{n^2} $	$ \theta = 0.1 $	$ l=0.001 $	$ \beta_n = \frac{1}{n} $
	$ \gamma = 0.99 $	$ \mu = 0.99 $
Thong Alg. (1)	$ \epsilon_n = \frac{1}{(n + 1)^2} $	$ \theta = 0.1 $	$ \beta_n = \frac{1}{n + 1} $	$ \lambda = \frac{1}{1.01L} $
Thong Alg. (2)	$ l=0.001 $	$ \gamma = 0.99 $	$ \mu = 0.99 $
Thong Alg. (3)	$ \alpha_n = \frac{1}{n + 1} $	$ l=0.001 $	$ \gamma = 0.99 $	$ \mu = 0.99 $
Gibali Alg.	$ \alpha_n = \frac{1}{n + 1} $	$ l=0.001 $	$ \gamma = 0.99 $	$ \mu = 0.99 $

	$ N=10 $		$ N=20 $		$ N=30 $		$ N=40 $
$ k=20 $	Iter.	Time	Iter.	Time	Iter.	Time	Iter.	Time
Proposed Alg.	3	1.3843	3	1.7672	3	1.7564	4	2.2017
Thong Alg. (1)	76	1.2902	139	2.7111	111	2.1715	232	37.7743
Thong Alg. (2)	2136	36.6812	1561	30.7776	1370	31.8672	1160	4.0453
Thong Alg. (3)	86	1.1655	152	2.3615	148	2.4878	178	29.0789
Gibali Alg.	150	12.0085	235	20.3243	319	41.0421	315	4.2520
	$ N=10 $		$ N=20 $		$ N=30 $		$ N=40 $
$ k=30 $	Iter.	Time	Iter.	Time	Iter.	Time	Iter.	Time
Proposed Alg.	3	1.3819	3	1.7834	3	1.6555	3	1.7517
Thong Alg. (1)	72	1.1548	142	2.6436	136	2.888	207	4.4416
Thong Alg. (2)	1771	30.2921	1325	28.4023	1132	28.6053	920	26.5714
Thong Alg. (3)	101	1.5058	90	1.4923	156	2.9515	162	3.6149
Gibali Alg.	203	17.1568	255	30.849	282	31.6244	303	35.2953

$ (N, k) $		$ \gamma = 0.1 $	$ \gamma = 0.5 $	$ \gamma = 0.7 $	$ \gamma = 0.99 $
$ (20, 10) $	No. of Iterations	3	3	3	3
	CPU (Time)	1.6337	1.4830	1.4773	1.3843
$ (20, 20) $	No. of Iterations	3	3	4	3
	CPU (Time)	1.4606	1.4876	2.3980	1.7672
$ (20, 30) $	No. of Iterations	4	5	4	3
	CPU (Time)	1.8664	0.85257	1.7597	1.7564
$ (20, 40) $	No. of Iterations	4	3	4	4
	CPU (Time)	1.6573	1.5935	1.8008	2.2017
$ (30, 10) $	No. of Iterations	3	3	3	3
	CPU (Time)	1.3060	1.3376	1.4359	1.3819
$ (30, 20) $	No. of Iterations	3	4	3	3
	CPU (Time)	1.4630	1.7306	1.5115	1.7834
$ (30, 30) $	No. of Iterations	4	3	4	3
	CPU (Time)	1.7102	1.6399	1.7931	1.6555
$ (30, 40) $	No. of Iterations	5	3	4	3
	CPU (Time)	2.7099	1.6589	2.3287	1.7517

$ (N, k) $		$ \mu = 0.1 $	$ \mu = 0.5 $	$ \mu = 0.7 $	$ \mu = 0.99 $
$ (20, 10) $	No. of Iterations	3	3	4	3
	CPU (Time)	1.4409	1.4632	2.6888	1.3843
$ (20, 20) $	No. of Iterations	3	3	3	3
	CPU (Time)	1.5248	1.4840	1.5217	1.7672
$ (20, 30) $	No. of Iterations	4	3	5	3
	CPU (Time)	1.9571	1.5852	1.9322	1.7564
$ (20, 40) $	No. of Iterations	6	4	5	4
	CPU (Time)	3.0365	1.8605	2.0718	2.2017
$ (30, 10) $	No. of Iterations	3	3	3	3
	CPU (Time)	1.3524	1.3416	1.3648	1.3819
$ (30, 20) $	No. of Iterations	3	3	4	3
	CPU (Time)	1.5265	1.5336	1.6929	1.7834
$ (30, 30) $	No. of Iterations	5	5	3	3
	CPU (Time)	2.9525	2.0816	1.6424	1.6555
$ (30, 40) $	No. of Iterations	3	4	8	3
	CPU (Time)	1.6958	2.0833	4.5199	1.7517

Proposed Alg.	$ \epsilon_n = \frac{1}{(n + 1)^2} $	$ \theta = 0.5 $	$ l=0.01 $	$ \beta_n = \frac{1}{n + 1} $	$ \gamma = 0.99 $	$ \mu = 0.99 $
Gibali Alg.	$ \alpha_n = \frac{1}{1 + n} $	$ l=0.01 $	$ \gamma = 0.99 $	$ \mu = 0.99 $

	No. of Iterations		CPU Time
	Prop. Alg.	Gibali Alg.	Prop. Alg.	Gibali Alg.
Case I	17	1712	0.001243	0.1244
Case II	17	1708	0.001518	0.1248
Case III	17	1713	0.001261	0.1276
Case IV	17	1729	0.001202	0.1297
Case V	17	1715	0.001272	0.1258
Case VI	18	1835	0.001339	0.1564

[1]	Grace Nnennaya Ogwo, Chinedu Izuchukwu, Oluwatosin Temitope Mewomo. A modified extragradient algorithm for a certain class of split pseudo-monotone variational inequality problem. Numerical Algebra, Control and Optimization, 2022, 12 (2) : 373-393. doi: 10.3934/naco.2021011
[2]	Shaotao Hu, Yuanheng Wang, Bing Tan, Fenghui Wang. Inertial iterative method for solving variational inequality problems of pseudo-monotone operators and fixed point problems of nonexpansive mappings in Hilbert spaces. Journal of Industrial and Management Optimization, 2022 doi: 10.3934/jimo.2022060
[3]	Lateef Olakunle Jolaoso, Maggie Aphane. Bregman subgradient extragradient method with monotone self-adjustment stepsize for solving pseudo-monotone variational inequalities and fixed point problems. Journal of Industrial and Management Optimization, 2022, 18 (2) : 773-794. doi: 10.3934/jimo.2020178
[4]	Yekini Shehu, Olaniyi Iyiola. On a modified extragradient method for variational inequality problem with application to industrial electricity production. Journal of Industrial and Management Optimization, 2019, 15 (1) : 319-342. doi: 10.3934/jimo.2018045
[5]	Augusto Visintin. Weak structural stability of pseudo-monotone equations. Discrete and Continuous Dynamical Systems, 2015, 35 (6) : 2763-2796. doi: 10.3934/dcds.2015.35.2763
[6]	Augusto VisintiN. On the variational representation of monotone operators. Discrete and Continuous Dynamical Systems - S, 2017, 10 (4) : 909-918. doi: 10.3934/dcdss.2017046
[7]	Zhili Ge, Gang Qian, Deren Han. Global convergence of an inexact operator splitting method for monotone variational inequalities. Journal of Industrial and Management Optimization, 2011, 7 (4) : 1013-1026. doi: 10.3934/jimo.2011.7.1013
[8]	A. C. Eberhard, J-P. Crouzeix. Existence of closed graph, maximal, cyclic pseudo-monotone relations and revealed preference theory. Journal of Industrial and Management Optimization, 2007, 3 (2) : 233-255. doi: 10.3934/jimo.2007.3.233
[9]	Françoise Demengel. Ergodic pairs for degenerate pseudo Pucci's fully nonlinear operators. Discrete and Continuous Dynamical Systems, 2021, 41 (7) : 3465-3488. doi: 10.3934/dcds.2021004
[10]	Anne-Laure Bessoud. A variational convergence for bifunctionals. Application to a model of strong junction. Discrete and Continuous Dynamical Systems - S, 2012, 5 (3) : 399-417. doi: 10.3934/dcdss.2012.5.399
[11]	Frank Jochmann. A variational inequality in Bean's model for superconductors with displacement current. Discrete and Continuous Dynamical Systems, 2009, 25 (2) : 545-565. doi: 10.3934/dcds.2009.25.545
[12]	Pierdomenico Pepe. A nonlinear version of Halanay's inequality for the uniform convergence to the origin. Mathematical Control and Related Fields, 2021 doi: 10.3934/mcrf.2021045
[13]	Li Wang, Yang Li, Liwei Zhang. A differential equation method for solving box constrained variational inequality problems. Journal of Industrial and Management Optimization, 2011, 7 (1) : 183-198. doi: 10.3934/jimo.2011.7.183
[14]	Hui-Qiang Ma, Nan-Jing Huang. Neural network smoothing approximation method for stochastic variational inequality problems. Journal of Industrial and Management Optimization, 2015, 11 (2) : 645-660. doi: 10.3934/jimo.2015.11.645
[15]	Ming Chen, Chongchao Huang. A power penalty method for a class of linearly constrained variational inequality. Journal of Industrial and Management Optimization, 2018, 14 (4) : 1381-1396. doi: 10.3934/jimo.2018012
[16]	Walter Allegretto, Yanping Lin, Shuqing Ma. On the box method for a non-local parabolic variational inequality. Discrete and Continuous Dynamical Systems - B, 2001, 1 (1) : 71-88. doi: 10.3934/dcdsb.2001.1.71
[17]	Shuang Chen, Li-Ping Pang, Dan Li. An inexact semismooth Newton method for variational inequality with symmetric cone constraints. Journal of Industrial and Management Optimization, 2015, 11 (3) : 733-746. doi: 10.3934/jimo.2015.11.733
[18]	Chibueze Christian Okeke, Abdulmalik Usman Bello, Lateef Olakunle Jolaoso, Kingsley Chimuanya Ukandu. Inertial method for split null point problems with pseudomonotone variational inequality problems. Numerical Algebra, Control and Optimization, 2021 doi: 10.3934/naco.2021037
[19]	Yarui Duan, Pengcheng Wu, Yuying Zhou. Penalty approximation method for a double obstacle quasilinear parabolic variational inequality problem. Journal of Industrial and Management Optimization, 2022 doi: 10.3934/jimo.2022017
[20]	Weijun Zhou, Youhua Zhou. On the strong convergence of a modified Hestenes-Stiefel method for nonconvex optimization. Journal of Industrial and Management Optimization, 2013, 9 (4) : 893-899. doi: 10.3934/jimo.2013.9.893

American Institute of Mathematical Sciences

Inertial Tseng's extragradient method for solving variational inequality problems of pseudo-monotone and non-Lipschitz operators

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Tools

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors