doi: 10.3934/naco.2021004

A new inertial-projection algorithm for approximating common solution of variational inequality and fixed point problems of multivalued mappings

School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Durban, South Africa

* Corresponding author: Oluwatosin Temitope Mewomo

Received  May 2020 Revised  January 2021 Published  February 2021

Fund Project: The third author is supported by International Mathematical Union (IMU) Breakout Graduate Fellowship and the fourth author is supported by National Research Foundation (NRF), South Africa, grant 119903

In this paper, we present a new modified self-adaptive inertial subgradient extragradient algorithm in which the two projections are made onto some half spaces. Moreover, under mild conditions, we obtain a strong convergence of the sequence generated by our proposed algorithm for approximating a common solution of variational inequality problem and common fixed point of a finite family of demicontractive mappings in a real Hilbert space. The main advantages of our algorithm are: strong convergence result obtained without prior knowledge of the Lipschitz constant of the related monotone operator, the two projections made onto some half-spaces and the inertial technique which speeds up rate of convergence. Finally, we present an application and a numerical example to illustrate the usefulness and applicability of our algorithm.

Citation: Abd-semii Oluwatosin-Enitan Owolabi, Timilehin Opeyemi Alakoya, Adeolu Taiwo, Oluwatosin Temitope Mewomo. A new inertial-projection algorithm for approximating common solution of variational inequality and fixed point problems of multivalued mappings. Numerical Algebra, Control & Optimization, doi: 10.3934/naco.2021004
References:
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H. A. Abass, K. O. Aremu, L. O. Jolaoso and O. T. Mewomo, An inertial forward-backward splitting method for approximating solutions of certain optimization problems, J. Nonlinear Funct. Anal., 2020 (2020), Art. ID 6, 20 pp. doi: 10.1287/moor.20.2.449.  Google Scholar

[2]

T. O. Alakoya, L. O. Jolaoso and O. T. Mewomo, A self adaptive inertial algorithm for solving split variational inclusion and fixed point problems with applications, J. Ind. Manag. Optim., 2020. doi: 10.3934/jimo.2020152.  Google Scholar

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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, 2020. doi: 10.1080/02331934.2020.1723586.  Google Scholar

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T. O. AlakoyaL. O. Jolaoso and O. T. Mewomo, Two modifications of the inertial Tseng extragradient method with self-adaptive step size for solving monotone variational inequality problems, Demonstr. Math., 53 (2020), 208-224.  doi: 10.1515/dema-2020-0013.  Google Scholar

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T. O. Alakoya, A. Taiwo, O. T. Mewomo and Y. J. Cho, An iterative algorithm for solving variational inequality, generalized mixed equilibrium, convex minimization and zeros problems for a class of nonexpansive-type mappings, Ann. Univ. Ferrara Sez. VII Sci. Mat., 2021. doi: 10.1007/s11565-020-00354-2.  Google Scholar

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C. Izuchukwu, A. A. Mebawondu and O. T. Mewomo, A new method for solving split variational inequality problems without co-coerciveness, J. Fixed Point Theory Appl., 22 (2020), Art. No. 98, 23 pp. doi: 10.1007/s11784-020-00834-0.  Google Scholar

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C. Izuchukwu, G. N. Ogwo and O. T. Mewomo, An inertial method for solving generalized split feasibility problems over the solution set of monotone variational inclusions, Optimization, 2020. doi: 10.1080/02331934.2020.1808648.  Google Scholar

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C. IzuchukwuC. C. Okeke and O. T. Mewomo, Systems of variational inequalities and multiple-set split equality fixed-point problems for countable families of multivalued type-one mappings of the demicontractive type, Ukra?n. Mat. Zh., 71 (2019), 1480-1501.   Google Scholar

[22]

C. IzuchukwuG. C. Ugwunnadi and O. T. Mewomo, Proximal-type algorithms for split minimization problem in p-uniformly convex metric space, Numer. Algorithms, 82 (2019), 909-935.  doi: 10.1007/s11075-018-0633-9.  Google Scholar

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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, 2020. doi: 10.1080/02331934.2020.1716752.  Google Scholar

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L. O. Jolaoso, K. O. Oyewole, K. O. Aremu and O. T. Mewomo, A new efficient algorithm for finding common fixed points of multivalued demicontractive mappings and solutions of split generalized equilibrium problems in Hilbert spaces, Int. J. Comput. Math., 2020. doi: 10.1080/00207160.2020.1856823.  Google Scholar

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L. O. JolaosoA. TaiwoT. O. Alakoya and O. T. Mewomo, Strong convergence theorem for solving pseudo-monotone variational inequality problem using projection method in a reflexive Banach space, J. Optim. Theory Appl., 185 (2020), 744-766.  doi: 10.1007/s10957-020-01672-3.  Google Scholar

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S. H. Khan, T. O. Alakoya and O. T. Mewomo, Relaxed projection methods with self-adaptive step size for solving variational inequality and fixed point problems for an infinite family of multivalued relatively nonexpansive mappings in Banach spaces, Math. Comput. Appl., 25 (2020), Art. 54. doi: 10.3390/mca25030054.  Google Scholar

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show all references

References:
[1]

H. A. Abass, K. O. Aremu, L. O. Jolaoso and O. T. Mewomo, An inertial forward-backward splitting method for approximating solutions of certain optimization problems, J. Nonlinear Funct. Anal., 2020 (2020), Art. ID 6, 20 pp. doi: 10.1287/moor.20.2.449.  Google Scholar

[2]

T. O. Alakoya, L. O. Jolaoso and O. T. Mewomo, A self adaptive inertial algorithm for solving split variational inclusion and fixed point problems with applications, J. Ind. Manag. Optim., 2020. doi: 10.3934/jimo.2020152.  Google Scholar

[3]

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, 2020. doi: 10.1080/02331934.2020.1723586.  Google Scholar

[4]

T. O. AlakoyaL. O. Jolaoso and O. T. Mewomo, Two modifications of the inertial Tseng extragradient method with self-adaptive step size for solving monotone variational inequality problems, Demonstr. Math., 53 (2020), 208-224.  doi: 10.1515/dema-2020-0013.  Google Scholar

[5]

T. O. Alakoya, A. Taiwo, O. T. Mewomo and Y. J. Cho, An iterative algorithm for solving variational inequality, generalized mixed equilibrium, convex minimization and zeros problems for a class of nonexpansive-type mappings, Ann. Univ. Ferrara Sez. VII Sci. Mat., 2021. doi: 10.1007/s11565-020-00354-2.  Google Scholar

[6]

K. O. AremuH. A. AbassC. Izuchukwu and O. T. Mewomo, A viscosity-type algorithm for an infinitely countable family of $(f, g)$-generalized k-strictly pseudononspreading mappings in CAT(0) spaces, Analysis, 40 (2020), 19-37.  doi: 10.1515/anly-2018-0078.  Google Scholar

[7]

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[8]

H. H. Bauschke and P. L. Combettes, A weak-to-strong convergence principle for Fejer-monotone method in Hilbert spaces, Math. Oper. Res., 26 (2001), 248-264.  doi: 10.1287/moor.26.2.248.10558.  Google Scholar

[9]

R. I. Bot and E. R. Csetnek, A hybrid proximal extragradient algorithm with inertial effects, Numer. Funct. Anal. Optim., 36 (2015), 951-963.  doi: 10.1080/01630563.2015.1042113.  Google Scholar

[10]

Y. CensorA. Gibali and S. Reich, The subgradient extragradient method for solving variational inequalities in Hilbert spaces, J. Optim. Theory Appl., 148 (2011), 318-335.  doi: 10.1007/s10957-010-9757-3.  Google Scholar

[11]

L. Q. DongY. Y. Lu and J. Yang, The extragradient algorithm with inertial effects for solving the variational inequality, Optimization, 65 (2016), 2217-2226.  doi: 10.1080/02331934.2016.1239266.  Google Scholar

[12]

G. Fichera, Sul problema elastostatico di signorini con ambigue condizioni al contorno, Atti Accad. Naz. Lincei, Ⅷ, Ser., Rend., Cl. Sci. Fis. Mat. Nat., 34 (1963), 138-142.   Google Scholar

[13]

A. Gibali, L. O. Jolaoso, O. T. Mewomo and A. Taiwo, Fast and simple Bregman projection methods for solving variational inequalities and related problems in Banach spaces, Results Math., 75 (2020), Art. No. 175. doi: 10.1007/s00025-020-01306-0.  Google Scholar

[14]

K. Goebel and S. Reich, Uniform convexity, hyperbolic geometry, and nonexpansive mappings, Marcel Dekker, New York, 1984.  Google Scholar

[15]

D. HieuK. P. Anh and L. D. Muu, Modified extragradient-like algorithms with new stepsizes for variational inequalities, Comput. Optim. Appl., 73 (2019), 913-932.  doi: 10.1007/s10589-019-00093-x.  Google Scholar

[16]

D. V. Hieu, Y. J. Cho and Y-B. Xiao, Modified extragradient method for pseudomonotone variational inequalities in infinite dimensional Hilbert spaces, Vietnam J. Math., 2020. doi: 10.1007/s10013-020-00447-7.  Google Scholar

[17]

D. V. HieuP. K. Anh and L. D. Muu, An explicit extragradient algorithm for solving variational inequalities, J. Optim. Theory Appl., 158 (2020), 476-503.  doi: 10.1007/s10957-020-01661-6.  Google Scholar

[18]

D. V. Hieu, P. K. Anh and L. D. Muu, Strong convergence of subgradient extragradient method with regularization for solving variational inequalities, Optim. Eng., 2020. doi: 10.1007/s11081-020-09540-9.  Google Scholar

[19]

C. Izuchukwu, A. A. Mebawondu and O. T. Mewomo, A new method for solving split variational inequality problems without co-coerciveness, J. Fixed Point Theory Appl., 22 (2020), Art. No. 98, 23 pp. doi: 10.1007/s11784-020-00834-0.  Google Scholar

[20]

C. Izuchukwu, G. N. Ogwo and O. T. Mewomo, An inertial method for solving generalized split feasibility problems over the solution set of monotone variational inclusions, Optimization, 2020. doi: 10.1080/02331934.2020.1808648.  Google Scholar

[21]

C. IzuchukwuC. C. Okeke and O. T. Mewomo, Systems of variational inequalities and multiple-set split equality fixed-point problems for countable families of multivalued type-one mappings of the demicontractive type, Ukra?n. Mat. Zh., 71 (2019), 1480-1501.   Google Scholar

[22]

C. IzuchukwuG. C. Ugwunnadi and O. T. Mewomo, Proximal-type algorithms for split minimization problem in p-uniformly convex metric space, Numer. Algorithms, 82 (2019), 909-935.  doi: 10.1007/s11075-018-0633-9.  Google Scholar

[23]

L. O. JolaosoT. O. AlakoyaA. Taiwo and O. T. Mewomo, A parallel combination extragradient method with Armijo line searching for finding common solution of finite families of equilibrium and fixed point problems, Rend. Circ. Mat. Palermo Ⅱ, 69 (2020), 711-735.  doi: 10.1007/s12215-019-00431-2.  Google Scholar

[24]

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, 2020. doi: 10.1080/02331934.2020.1716752.  Google Scholar

[25]

L. O. Jolaoso, K. O. Oyewole, K. O. Aremu and O. T. Mewomo, A new efficient algorithm for finding common fixed points of multivalued demicontractive mappings and solutions of split generalized equilibrium problems in Hilbert spaces, Int. J. Comput. Math., 2020. doi: 10.1080/00207160.2020.1856823.  Google Scholar

[26]

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), Art. No. 38. doi: 10.1007/s40314-019-1014-2.  Google Scholar

[27]

L. O. JolaosoA. TaiwoT. O. Alakoya and O. T. Mewomo, Strong convergence theorem for solving pseudo-monotone variational inequality problem using projection method in a reflexive Banach space, J. Optim. Theory Appl., 185 (2020), 744-766.  doi: 10.1007/s10957-020-01672-3.  Google Scholar

[28]

G. KassayS. Reich and S. Sabach, Iterative methods for solving systems of variational inequalities in reflexive Banach spaces, SIAM J. Optim., 21 (2011), 1319-1344.  doi: 10.1137/110820002.  Google Scholar

[29]

S. H. Khan, T. O. Alakoya and O. T. Mewomo, Relaxed projection methods with self-adaptive step size for solving variational inequality and fixed point problems for an infinite family of multivalued relatively nonexpansive mappings in Banach spaces, Math. Comput. Appl., 25 (2020), Art. 54. doi: 10.3390/mca25030054.  Google Scholar

[30]

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

[31]

R. Kraikaew and S. Saejung, Strong convergence of the Halpern subgradient extragradient method for solving variational inequalities in Hilbert spaces, J. Optim. Theory Appl., 163 (2014), 399-412.  doi: 10.1007/s10957-013-0494-2.  Google Scholar

[32]

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.  Google Scholar

[33]

P. E. Maingé and M. L. Gbinddass, Convergence of one-step projected gradient methods for variational inequalities, J. Optim. Theory Appl., 171 (2016), 146-168.  doi: 10.1007/s10957-016-0972-4.  Google Scholar

[34]

Y. V. Malitsky, Projected reflected gradient methods for monotone variational inequalities, SIAM J. Optim., 25 (2015), 502-520.  doi: 10.1137/14097238X.  Google Scholar

[35]

G. Marino and H. K. Xu, Weak and strong convergence theorems for pseudo-contraction in Hilbert spaces, J. Math. Anal. Appl., 329 (2007), 336-346.  doi: 10.1016/j.jmaa.2006.06.055.  Google Scholar

[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 (2006), 1230-1241.  doi: 10.1137/050624315.  Google Scholar

[37]

N. Nadezhkina and W. Takahashi, Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings, J. Optim. Theory Appl., 128 (2006), 191-201.  doi: 10.1007/s10957-005-7564-z.  Google Scholar

[38]

C. C. OkekeC. Izuchukwu and O. T. Mewomo, Strong convergence results for convex minimization and monotone variational inclusion problems in Hilbert space, Rend. Circ. Mat. Palermo (2), 69 (2020), 675-693.  doi: 10.1007/s12215-019-00427-y.  Google Scholar

[39]

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Figure 1.  From Top to Bottom: Case 1 - Case 4
Table 1.  Numerical results
Alg. 3.1 Non-inertial
Case 1 CPU time (sec) 1.9051 2.1221
No of Iter. 4 6
Case 2 CPU time (sec) 1.8332 1.9000
No. of Iter. 4 6
Case 3 CPU time (sec) 1.9193 2.1708
No of Iter. 4 6
Case 4 CPU time (sec) 16.5580 24.9989
No of Iter. 4 5
Alg. 3.1 Non-inertial
Case 1 CPU time (sec) 1.9051 2.1221
No of Iter. 4 6
Case 2 CPU time (sec) 1.8332 1.9000
No. of Iter. 4 6
Case 3 CPU time (sec) 1.9193 2.1708
No of Iter. 4 6
Case 4 CPU time (sec) 16.5580 24.9989
No of Iter. 4 5
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