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January  2022, 18(1): 239-265. doi: 10.3934/jimo.2020152

A self adaptive inertial algorithm for solving split variational inclusion and fixed point problems with applications

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

* Corresponding author: Oluwatosin Temitope Mewomo

Received  November 2019 Revised  March 2020 Published  January 2022 Early access  October 2020

We propose a general iterative scheme with inertial term and self-adaptive stepsize for approximating a common solution of Split Variational Inclusion Problem (SVIP) and Fixed Point Problem (FPP) for a quasi-nonexpansive mapping in real Hilbert spaces. We prove that our iterative scheme converges strongly to a common solution of SVIP and FPP for a quasi-nonexpansive mapping, which is also a solution of a certain optimization problem related to a strongly positive bounded linear operator. We apply our proposed algorithm to the problem of finding an equilibrium point with minimal cost of production for a model in industrial electricity production. Numerical results are presented to demonstrate the efficiency of our algorithm in comparison with some other existing algorithms in the literature.

Citation: Timilehin Opeyemi Alakoya, Lateef Olakunle Jolaoso, Oluwatosin Temitope Mewomo. A self adaptive inertial algorithm for solving split variational inclusion and fixed point problems with applications. Journal of Industrial and Management Optimization, 2022, 18 (1) : 239-265. doi: 10.3934/jimo.2020152
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.

[2]

M. AbbasM. Al SharaniQ. H. AnsariO. S. Iyiola and Y. Shehu, Iterative methods for solving proximal split minimization problem, Numer. Algorithms, 78 (2018), 193-215.  doi: 10.1007/s11075-017-0372-3.

[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.

[4]

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.

[5]

J. P. Aubin, Optima and Equilibria: An Introduction to Nonlinear Analysis, Springer, 1993. doi: 10.1007/978-3-662-02959-6.

[6]

A. Beck and M. Teboulle, A fast iterative shrinkage-thresholding algorithm for linear inverse problem, SIAM J. Imaging Sci., 2 (2009), 183-202.  doi: 10.1137/080716542.

[7]

F. E. Browder, Semicontractive and semiaccretive nonlinear mappings in Banach spaces, Bull. Amer. Math. Soc., 74 (1968), 660-665.  doi: 10.1090/S0002-9904-1968-11983-4.

[8]

C. Byrne, Iterative oblique projection onto convex subsets and the split feasibility problem, Inverse Probl., 18 (2002), 441-453.  doi: 10.1088/0266-5611/18/2/310.

[9]

C. ByrneY. CensorA. Gibali and S. Reich, Weak and strong convergence of algorithms for the split common null point problem, J. Nonlinear Convex Anal., 13 (2012), 759-775. 

[10]

L. C. CengQ. H. Ansari and J. C. Yao, An extragradient method for solving split feasibility and fixed point problems, Comput. Math. Appl., 64 (2012), 633-642.  doi: 10.1016/j.camwa.2011.12.074.

[11]

Y. Censor and T. Elfving, A multiprojection algorithms using Bragman projection in a product space, Numer. Algorithms, 8 (1994), 221-239.  doi: 10.1007/BF02142692.

[12]

Y. CensorT. ElfvingN. Kopf and T. Bortfeld, The multiple-sets split feasibility problem and its applications for inverse problems, Inverse Probl., 21 (2005), 2071-2084.  doi: 10.1088/0266-5611/21/6/017.

[13]

L. C. CengQ. H. Ansari and J. C. Yao, An extragradient method for solving split feasibility and fixed point problems, Comput. Math. Appl., 64 (2012), 633-642.  doi: 10.1016/j.camwa.2011.12.074.

[14]

A. Chambolle and C. Dossal, On the convergence of the iterates of the "fast iterative shrinkage/thresholding algorithm", J. Optim. Theory Appl., 166 (2015), 968-982.  doi: 10.1007/s10957-015-0746-4.

[15]

R. H. ChanS. Ma and J. F. Jang, Inertial proximal ADMM for linearly constrained separable convex optimization, SIAM J. Imaging Sci., 8 (2015), 2239-2267.  doi: 10.1137/15100463X.

[16]

R. Glowinski and P. Le Tallec, Augmented Lagrangian and operator-splitting methods in nonlinear mechanics, SIAM, Philadelphia, 9 (1989). doi: 10.1137/1.9781611970838.

[17]

A. N. Iusem, On some properties of paramonotone operator, Convex Anal., 5 (1998), 269-278. 

[18]

C. IzuchukwuG. C. UgwunnadiO. T. MewomoA. R. Khan and M. Abbas, 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.

[19]

L. O. Jolaoso, T. O. Alakoya, A. 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, (2019). doi: 10.1007/s12215-019-00431-2.

[20]

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.

[21]

L. O. JolaosoF. U. Ogbuisi and O. T. Mewomo, An iterative method for solving minimization, variational inequality and fixed point problems in reflexive Banach spaces, Adv. Pure Appl. Math., 9 (2018), 167-184.  doi: 10.1515/apam-2017-0037.

[22]

L. O. JolaosoK. O. OyewoleC. C. Okeke and O. T. Mewomo, A unified algorithm for solving split generalized mixed equilibrium problem and fixed point of nonspreading mapping in Hilbert space, Demonstr. Math., 51 (2018), 211-232.  doi: 10.1515/dema-2018-0015.

[23]

L. O. JolaosoA. TaiwoT. O. Alakoya and O. T. Mewomo, A self adaptive inertial subgradient extragradient algorithm for variational inequality and common fixed point of multivalued mappings in Hilbert spaces, Demonstr. Math., 52 (2019), 183-203.  doi: 10.1515/dema-2019-0013.

[24]

Y. Kimura and S. Saejung, Strong convergence for a common fixed point of two different generalizations of cutter operators, Linear Nonlinear Anal., 1 (2015), 53-65. 

[25]

L. V. Long, D. V. Thong and V. T. Dung, New algorithms for the split variational inclusion problems and application to split feasibility problems, Optimizaton, (2019). doi: 10.1080/02331934.2019.1631821.

[26]

D. Lorenz and T. Pock, An inertial forward-backward algorithm for monotone inclusions, J. Math. Imaging Vision, 51 (2015), 311-325.  doi: 10.1007/s10851-014-0523-2.

[27]

P. E. Maingé, Approximation methods for common fixed points of nonexpansive mappings in Hilbert spaces, J. Math. Anal. Appl., 325 (2007), 469-479.  doi: 10.1016/j.jmaa.2005.12.066.

[28]

P. E. Maingé, The viscosity approximation process for quasi-nonexpansive mappings in Hilbert spaces, Comput. Math. Appl., 59 (2010), 74-79.  doi: 10.1016/j.camwa.2009.09.003.

[29]

G. Marino and H. K. Xu, A general iterative method for nonexpansive mapping in Hilbert spaces, J. Math. Anal. Appl., 318 (2006), 43-52.  doi: 10.1016/j.jmaa.2005.05.028.

[30]

A. Moudafi, Split monotone variational inclusions, J. Optim. Theory Appl., 150 (2011), 275-283.  doi: 10.1007/s10957-011-9814-6.

[31]

A. Moudafi, Viscosity approximation methods for fixed-points problems, J. Math. Anal. Appl., 241 (2000), 46-55.  doi: 10.1006/jmaa.1999.6615.

[32]

A. Moudafi and M. Oliny, Convergence of a splitting inertial proximal method for monotone operators, J. Comput. Appl. Math., 152 (2003), 447-454.  doi: 10.1016/S0377-0427(02)00906-8.

[33]

F. U. Ogbuisi and O. T. Mewomo, Convergence analysis of an inertial accelerated iterative algorithm for solving split variational inequality problem, Adv. Pure Appl. Math., 10 (2019), 339-353.  doi: 10.1515/apam-2017-0132.

[34]

F. U. Ogbuisi and O. T. Mewomo, Convergence analysis of common solution of certain nonlinear problems, Fixed Point Theory, 19 (2018), 335-358.  doi: 10.24193/fpt-ro.2018.1.26.

[35]

F. U. Ogbuisi and O. T. Mewomo, Iterative solution of split variational inclusion problem in real Banach space, Afr. Mat., 28 (2017), 295-309.  doi: 10.1007/s13370-016-0450-z.

[36]

G. N. OgwoC. IzuchukwuK. O. Aremu and O. T. Mewomo, A viscosity iterative algorithm for a family of monotone inclusion problems in an Hadamard space, Bull. Belg. Math. Soc. Simon Stevin, 27 (2020), 127-152.  doi: 10.36045/bbms/1590199308.

[37]

C. C. Okeke and O. T. Mewomo, On split equilibrium problem, variational inequality problem and fixed point problem for multivalued mappings, Ann. Acad. Rom. Sci. Ser. Math. Appl., 9 (2017), 255-280. 

[38]

P. Phairatchatniyom, P. Kumam, Y. J. Cho, W. Jirakitpuwapat and K. Sitthithakerngkiet, The modified inertial iterative algorithm for solving split variational inclusion problem for multi-valued quasi nonexpansive mappings with some applications, Mathematics, 7 (2019), 560. doi: 10.3390/math7060560.

[39]

B. T. Polyak, Some methods of speeding up the convergence of iterative methods, Zh. Vychisl. Mat. Mat. Fiz., 4 (1964), 1-17. 

[40]

S. Saejung and P. Yotkaew, Approximation of zeros of inverse strongly monotone operators in Banach spaces, Nonlinear Anal., 75 (2012), 742-750.  doi: 10.1016/j.na.2011.09.005.

[41]

Y. Shehu and D. Agbebaku, On split inclusion problem and fixed point problem for multi-valued mappings, Comput. Appl. Math., 37 (2018), 1807-1824.  doi: 10.1007/s40314-017-0426-0.

[42]

Y. Shehu and O. T. Mewomo, Further investigation into split common fixed point problem for demicontractive operators, Acta Math. Sin. (Engl. Ser.), 32 (2016), 1357-1376.  doi: 10.1007/s10114-016-5548-6.

[43]

A. Taiwo, L. O. Jolaoso and O. T. Mewomo, A modified Halpern algorithm for approximating a common solution of split equality convex minimization problem and fixed point problem in uniformly convex Banach spaces, Comput. Appl. Math., 38 (2019), Art. 77. doi: 10.1007/s40314-019-0841-5.

[44]

A. TaiwoL. O. Jolaoso and O. T. Mewomo, Parallel hybrid algorithm for solving pseudomonotone equilibrium and split common fixed point problems, Bull. Malays. Math. Sci. Soc., 43 (2020), 1893-1918.  doi: 10.1007/s40840-019-00781-1.

[45]

A. TaiwoL. O. Jolaoso and O. T. Mewomo, General alternative regularization method for solving split equality common fixed point problem for quasi-pseudocontractive mappings in Hilbert spaces, Ric. Mat., 69 (2020), 235-259.  doi: 10.1007/s11587-019-00460-0.

[46]

Y. Tang, Convergence analysis of a new iterative algorithm for solving split variational inclusion problems, J. Indus. Mgt Opt., 16 (2020), 945-964.  doi: 10.3934/jimo.2018187.

[47]

D. Van Hieu, Strong convergence of a new hybrid algorithm for fixed point problems and equilibrium problems, Math. Model. Anal., 24 (2019), 1-19.  doi: 10.3846/mma.2019.001.

[48]

R. WangkeereeK. Rattanaseeha and R. Wangkeeree, The general iterative methods for split variational inclusion problem and fixed point problem in Hilbert spaces, J. Comp. Anal. Appl., 25 (2018), 19-31. 

[49]

H. K. Xu, An iterative approach to quadratic optimization, J. Opt. Theory Appl., 116 (2003), 659-678.  doi: 10.1023/A:1023073621589.

[50]

H. K. Xu, Iterative algorithms for nonlinear operators, J. London Math. Soc., 66 (2002), 240-256.  doi: 10.1112/S0024610702003332.

[51]

Y. YaoM. PostolacheX. Qin and J.-C. Yao, Iterative algorithm for proximal split feasibility problem, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys., 80 (2018), 37-44. 

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.

[2]

M. AbbasM. Al SharaniQ. H. AnsariO. S. Iyiola and Y. Shehu, Iterative methods for solving proximal split minimization problem, Numer. Algorithms, 78 (2018), 193-215.  doi: 10.1007/s11075-017-0372-3.

[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.

[4]

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.

[5]

J. P. Aubin, Optima and Equilibria: An Introduction to Nonlinear Analysis, Springer, 1993. doi: 10.1007/978-3-662-02959-6.

[6]

A. Beck and M. Teboulle, A fast iterative shrinkage-thresholding algorithm for linear inverse problem, SIAM J. Imaging Sci., 2 (2009), 183-202.  doi: 10.1137/080716542.

[7]

F. E. Browder, Semicontractive and semiaccretive nonlinear mappings in Banach spaces, Bull. Amer. Math. Soc., 74 (1968), 660-665.  doi: 10.1090/S0002-9904-1968-11983-4.

[8]

C. Byrne, Iterative oblique projection onto convex subsets and the split feasibility problem, Inverse Probl., 18 (2002), 441-453.  doi: 10.1088/0266-5611/18/2/310.

[9]

C. ByrneY. CensorA. Gibali and S. Reich, Weak and strong convergence of algorithms for the split common null point problem, J. Nonlinear Convex Anal., 13 (2012), 759-775. 

[10]

L. C. CengQ. H. Ansari and J. C. Yao, An extragradient method for solving split feasibility and fixed point problems, Comput. Math. Appl., 64 (2012), 633-642.  doi: 10.1016/j.camwa.2011.12.074.

[11]

Y. Censor and T. Elfving, A multiprojection algorithms using Bragman projection in a product space, Numer. Algorithms, 8 (1994), 221-239.  doi: 10.1007/BF02142692.

[12]

Y. CensorT. ElfvingN. Kopf and T. Bortfeld, The multiple-sets split feasibility problem and its applications for inverse problems, Inverse Probl., 21 (2005), 2071-2084.  doi: 10.1088/0266-5611/21/6/017.

[13]

L. C. CengQ. H. Ansari and J. C. Yao, An extragradient method for solving split feasibility and fixed point problems, Comput. Math. Appl., 64 (2012), 633-642.  doi: 10.1016/j.camwa.2011.12.074.

[14]

A. Chambolle and C. Dossal, On the convergence of the iterates of the "fast iterative shrinkage/thresholding algorithm", J. Optim. Theory Appl., 166 (2015), 968-982.  doi: 10.1007/s10957-015-0746-4.

[15]

R. H. ChanS. Ma and J. F. Jang, Inertial proximal ADMM for linearly constrained separable convex optimization, SIAM J. Imaging Sci., 8 (2015), 2239-2267.  doi: 10.1137/15100463X.

[16]

R. Glowinski and P. Le Tallec, Augmented Lagrangian and operator-splitting methods in nonlinear mechanics, SIAM, Philadelphia, 9 (1989). doi: 10.1137/1.9781611970838.

[17]

A. N. Iusem, On some properties of paramonotone operator, Convex Anal., 5 (1998), 269-278. 

[18]

C. IzuchukwuG. C. UgwunnadiO. T. MewomoA. R. Khan and M. Abbas, 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.

[19]

L. O. Jolaoso, T. O. Alakoya, A. 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, (2019). doi: 10.1007/s12215-019-00431-2.

[20]

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.

[21]

L. O. JolaosoF. U. Ogbuisi and O. T. Mewomo, An iterative method for solving minimization, variational inequality and fixed point problems in reflexive Banach spaces, Adv. Pure Appl. Math., 9 (2018), 167-184.  doi: 10.1515/apam-2017-0037.

[22]

L. O. JolaosoK. O. OyewoleC. C. Okeke and O. T. Mewomo, A unified algorithm for solving split generalized mixed equilibrium problem and fixed point of nonspreading mapping in Hilbert space, Demonstr. Math., 51 (2018), 211-232.  doi: 10.1515/dema-2018-0015.

[23]

L. O. JolaosoA. TaiwoT. O. Alakoya and O. T. Mewomo, A self adaptive inertial subgradient extragradient algorithm for variational inequality and common fixed point of multivalued mappings in Hilbert spaces, Demonstr. Math., 52 (2019), 183-203.  doi: 10.1515/dema-2019-0013.

[24]

Y. Kimura and S. Saejung, Strong convergence for a common fixed point of two different generalizations of cutter operators, Linear Nonlinear Anal., 1 (2015), 53-65. 

[25]

L. V. Long, D. V. Thong and V. T. Dung, New algorithms for the split variational inclusion problems and application to split feasibility problems, Optimizaton, (2019). doi: 10.1080/02331934.2019.1631821.

[26]

D. Lorenz and T. Pock, An inertial forward-backward algorithm for monotone inclusions, J. Math. Imaging Vision, 51 (2015), 311-325.  doi: 10.1007/s10851-014-0523-2.

[27]

P. E. Maingé, Approximation methods for common fixed points of nonexpansive mappings in Hilbert spaces, J. Math. Anal. Appl., 325 (2007), 469-479.  doi: 10.1016/j.jmaa.2005.12.066.

[28]

P. E. Maingé, The viscosity approximation process for quasi-nonexpansive mappings in Hilbert spaces, Comput. Math. Appl., 59 (2010), 74-79.  doi: 10.1016/j.camwa.2009.09.003.

[29]

G. Marino and H. K. Xu, A general iterative method for nonexpansive mapping in Hilbert spaces, J. Math. Anal. Appl., 318 (2006), 43-52.  doi: 10.1016/j.jmaa.2005.05.028.

[30]

A. Moudafi, Split monotone variational inclusions, J. Optim. Theory Appl., 150 (2011), 275-283.  doi: 10.1007/s10957-011-9814-6.

[31]

A. Moudafi, Viscosity approximation methods for fixed-points problems, J. Math. Anal. Appl., 241 (2000), 46-55.  doi: 10.1006/jmaa.1999.6615.

[32]

A. Moudafi and M. Oliny, Convergence of a splitting inertial proximal method for monotone operators, J. Comput. Appl. Math., 152 (2003), 447-454.  doi: 10.1016/S0377-0427(02)00906-8.

[33]

F. U. Ogbuisi and O. T. Mewomo, Convergence analysis of an inertial accelerated iterative algorithm for solving split variational inequality problem, Adv. Pure Appl. Math., 10 (2019), 339-353.  doi: 10.1515/apam-2017-0132.

[34]

F. U. Ogbuisi and O. T. Mewomo, Convergence analysis of common solution of certain nonlinear problems, Fixed Point Theory, 19 (2018), 335-358.  doi: 10.24193/fpt-ro.2018.1.26.

[35]

F. U. Ogbuisi and O. T. Mewomo, Iterative solution of split variational inclusion problem in real Banach space, Afr. Mat., 28 (2017), 295-309.  doi: 10.1007/s13370-016-0450-z.

[36]

G. N. OgwoC. IzuchukwuK. O. Aremu and O. T. Mewomo, A viscosity iterative algorithm for a family of monotone inclusion problems in an Hadamard space, Bull. Belg. Math. Soc. Simon Stevin, 27 (2020), 127-152.  doi: 10.36045/bbms/1590199308.

[37]

C. C. Okeke and O. T. Mewomo, On split equilibrium problem, variational inequality problem and fixed point problem for multivalued mappings, Ann. Acad. Rom. Sci. Ser. Math. Appl., 9 (2017), 255-280. 

[38]

P. Phairatchatniyom, P. Kumam, Y. J. Cho, W. Jirakitpuwapat and K. Sitthithakerngkiet, The modified inertial iterative algorithm for solving split variational inclusion problem for multi-valued quasi nonexpansive mappings with some applications, Mathematics, 7 (2019), 560. doi: 10.3390/math7060560.

[39]

B. T. Polyak, Some methods of speeding up the convergence of iterative methods, Zh. Vychisl. Mat. Mat. Fiz., 4 (1964), 1-17. 

[40]

S. Saejung and P. Yotkaew, Approximation of zeros of inverse strongly monotone operators in Banach spaces, Nonlinear Anal., 75 (2012), 742-750.  doi: 10.1016/j.na.2011.09.005.

[41]

Y. Shehu and D. Agbebaku, On split inclusion problem and fixed point problem for multi-valued mappings, Comput. Appl. Math., 37 (2018), 1807-1824.  doi: 10.1007/s40314-017-0426-0.

[42]

Y. Shehu and O. T. Mewomo, Further investigation into split common fixed point problem for demicontractive operators, Acta Math. Sin. (Engl. Ser.), 32 (2016), 1357-1376.  doi: 10.1007/s10114-016-5548-6.

[43]

A. Taiwo, L. O. Jolaoso and O. T. Mewomo, A modified Halpern algorithm for approximating a common solution of split equality convex minimization problem and fixed point problem in uniformly convex Banach spaces, Comput. Appl. Math., 38 (2019), Art. 77. doi: 10.1007/s40314-019-0841-5.

[44]

A. TaiwoL. O. Jolaoso and O. T. Mewomo, Parallel hybrid algorithm for solving pseudomonotone equilibrium and split common fixed point problems, Bull. Malays. Math. Sci. Soc., 43 (2020), 1893-1918.  doi: 10.1007/s40840-019-00781-1.

[45]

A. TaiwoL. O. Jolaoso and O. T. Mewomo, General alternative regularization method for solving split equality common fixed point problem for quasi-pseudocontractive mappings in Hilbert spaces, Ric. Mat., 69 (2020), 235-259.  doi: 10.1007/s11587-019-00460-0.

[46]

Y. Tang, Convergence analysis of a new iterative algorithm for solving split variational inclusion problems, J. Indus. Mgt Opt., 16 (2020), 945-964.  doi: 10.3934/jimo.2018187.

[47]

D. Van Hieu, Strong convergence of a new hybrid algorithm for fixed point problems and equilibrium problems, Math. Model. Anal., 24 (2019), 1-19.  doi: 10.3846/mma.2019.001.

[48]

R. WangkeereeK. Rattanaseeha and R. Wangkeeree, The general iterative methods for split variational inclusion problem and fixed point problem in Hilbert spaces, J. Comp. Anal. Appl., 25 (2018), 19-31. 

[49]

H. K. Xu, An iterative approach to quadratic optimization, J. Opt. Theory Appl., 116 (2003), 659-678.  doi: 10.1023/A:1023073621589.

[50]

H. K. Xu, Iterative algorithms for nonlinear operators, J. London Math. Soc., 66 (2002), 240-256.  doi: 10.1112/S0024610702003332.

[51]

Y. YaoM. PostolacheX. Qin and J.-C. Yao, Iterative algorithm for proximal split feasibility problem, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys., 80 (2018), 37-44. 

Figure 1.  Example 5.1, Top Left: $ N = 50 $; Top Left: $ N = 100 $; Bottom Left: $ N = 500 $; Bottom Right: $ N = 1000 $
Figure 2.  Example 5.2, Top Left: Case I; Top Left: Case II; Bottom Left: Case III; Bottom Right: Case IV
Figure 3.  Example 5.3, Top Left: Choice (i); Top Left: Choice (ii); Bottom Left: Choice (iii); Bottom Right: Choice (iv)
Table 1.  Numerical results for Example 5.1
No of Iteration CPU time (sec)
$ N= 50 $ 19 0.0289
$ N=100 $ 19 0.0386
$ N=500 $ 41 0.1386
$ N=1000 $ 138 0.3523
No of Iteration CPU time (sec)
$ N= 50 $ 19 0.0289
$ N=100 $ 19 0.0386
$ N=500 $ 41 0.1386
$ N=1000 $ 138 0.3523
Table 2.  Numerical results for Example 5.2
Algorithm 3.1 Algorithm 1.1 Algorithm 1.2
Case I CPU time (sec) 0.0021 0.0071 0.0036
$ x_0 = 1, x_1 = 0.5 $ No of Iter. 8 22 16
Case II CPU time (sec) 0.0021 0.0041 0.0047
$ x_0 = -0.5, x_1 = 2 $ No. of Iter. 9 25 17
Case III CPU time (sec) 0.0044 0.0532 0.0095
$ x_0 = 5, x_1 = 10 $ No of Iter. 10 28 19
Case IV CPU time (sec) 0.0062 0.0589 0.0071
$ x_0 = -5, x_1 = 2 $ No of Iter. 10 27 19
Algorithm 3.1 Algorithm 1.1 Algorithm 1.2
Case I CPU time (sec) 0.0021 0.0071 0.0036
$ x_0 = 1, x_1 = 0.5 $ No of Iter. 8 22 16
Case II CPU time (sec) 0.0021 0.0041 0.0047
$ x_0 = -0.5, x_1 = 2 $ No. of Iter. 9 25 17
Case III CPU time (sec) 0.0044 0.0532 0.0095
$ x_0 = 5, x_1 = 10 $ No of Iter. 10 28 19
Case IV CPU time (sec) 0.0062 0.0589 0.0071
$ x_0 = -5, x_1 = 2 $ No of Iter. 10 27 19
Table 3.  Numerical results for Example 5.3
Algorithm 3.1 Algorithm 1.1
Choice (i) CPU time (sec) 1.7859 5.1231
No of Iter. 11 23
Choice (ii) CPU time (sec) 1.4997 13.3981
No. of Iter. 13 27
Choice (iii) CPU time (sec) 2.6789 9.1093
No of Iter. 7 12
Choice (iv) CPU time (sec) 6.3222 24.5622
No of Iter. 11 24
Algorithm 3.1 Algorithm 1.1
Choice (i) CPU time (sec) 1.7859 5.1231
No of Iter. 11 23
Choice (ii) CPU time (sec) 1.4997 13.3981
No. of Iter. 13 27
Choice (iii) CPU time (sec) 2.6789 9.1093
No of Iter. 7 12
Choice (iv) CPU time (sec) 6.3222 24.5622
No of Iter. 11 24
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