Inertial Mann-Krasnoselskii algorithm with self adaptive stepsize for split variational inclusion problem and paramonotone equilibria
Abstract
In this paper, we introduce a Mann-Krasnoselskii algorithm of inertial form for approximating a common solution of Spit Variational Inclusion Problem (SVIP) and Equilibrium Problem (EP) with paramonotone bifunction in real Hilbert spaces. Motivated by the self-adaptive technique, we incorporate the inertial technique to accelerate the convergence of the proposed method. Under standard and mild assumptions such as monotonicity and lower semicontinuity of the SVIP and EP associated mappings, we establish the strong convergence of the iterative algorithm. Some applications and numerical experiments are presented to illustrate the performance and behaviour of our method as well as comparing it with some related methods in the literature. Our results improve and generalize many existing results in this direction.
Keyword : split variational inclusion, equilibrium problem, pseudomonotonicity, self adaptive stepsize
How to Cite
Jolaoso, L. O., Mebawondu, A. A., & Mewomo, O. T. (2022). Inertial Mann-Krasnoselskii algorithm with self adaptive stepsize for split variational inclusion problem and paramonotone equilibria. Mathematical Modelling and Analysis, 27(2), 179–198. https://doi.org/10.3846/mma.2022.13949
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
M. Abbas, M. Al Sharani, Q.H. Ansari, G.S. Iyiola and Y. Shehu. Iterative methods for solving proximal split minimization problems. Numerical Algorithms, 78:193–215, 2018. https://doi.org/10.1007/s11075-017-0372-3
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. Journal of Industrial & Management Optimization, 18(1):239–265, 2022. https://doi.org/10.3934/jimo.2020152
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. Annali Dell’Universita’ Di Ferrara, 67(1):1–31, 2021. https://doi.org/10.1007/s11565-020-00354-2
P.N. Anh and L.D. Muu. A hybrid subgradient algorithm for nonexpansive mapping and equilibrium problems. Optimization Letters, 8:727–738, 2014.
E. Blum. From optimization and variational inequalities to equilibrium problems. The Mathematics Student, 63(1-4):123–145, 2014.
C.S. Chuang. Strong convergence theorems for the split variational inclusion problem in Hilbert spaces. Fixed Point Theory and Applications, 350(2013), 2013. https://doi.org/10.1186/1687-1812-2013-350
C.S. Chuang. Algorithms with new parameter conditions for split variational inclusion problems in Hilbert spaces with application to split feasibility problem. Optimization, 65:859–876, 2016. https://doi.org/10.1080/02331934.2015.1072715
K. Goebel and W.A. Wirk. Topics in metric fixed point theory. Cambridge studies in Advanced Mathematics, 28, Cambridge University Press, Cambridge, 1990.
H. Iiduka. Acceleration method for convex optimization over the fixed point set of a nonexpansive mappings. Mathematical Programming, 149:131–165, 2015. https://doi.org/10.1007/s10107-013-0741-1
C. Izuchukwu, A.A. Mebawondu and O.T. Mewomo. A new method for solving split variational inequality problems without co-coerciveness. Journal of Fixed Point Theory and Applications, 22(98), 2020. https://doi.org/10.1007/s11784-020-00834-0
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(2):387–412, 2021. https://doi.org/10.1080/02331934.2020.1716752
S. Kesornprom and P. Cholamjiak. Proximal type algorithms involving linesearch and inertial technique for split variational inclusion problem in Hilbert spaces with applications. Optimization, 68(12):2365–2391, 2019. https://doi.org/10.1080/02331934.2019.1638389
P.E. Maing´e. A hybrid extragradient-viscosity method for monotone operators and fixed point problems. SIAM Journal on Control and Optimization, 47:1499– 1515, 2008. https://doi.org/10.1137/060675319
A. Moudafi. Split monotone variational inclusions. Journal of Optimization Theory and Applications, 150:275–283, 2011. https://doi.org/10.1007/s10957-011-9814-6
A. Moudafi and B.S. Thakur. Solving proximal split feasibility problems without prior knowledge of operator norms. Optimization Letters, 8:2099–2110, 2014. https://doi.org/10.1007/s11590-013-0708-4
G.N. Ogwo, C. Izuchukwu and O.T. Mewomo. Inertial methods for finding minimum-norm solutions of the split variational inequality problem beyond monotonicity. Numerical Algorithms volume, 88:1419–1456, 2021. https://doi.org/10.1007/s11075-021-01081-1
M.A. Olona, T.O. Alakoya, A.O.-E. Owolabi and O.T. Mewomo. Inertial algorithm for solving equilibrium, variational inclusion and fixed point problems for an infinite family of strictly pseudocontractive mappings. Journal of Nonlinear Functional Analysis, 2021:1–21, 2021. https://doi.org/10.23952/jnfa.2021.10
M.A. Olona, T.O. Alakoya, A.O.-E. Owolabi and O.T. Mewomo. Inertial shrinking projection algorithm with self-adaptive step size for split generalized equilibrium and fixed point problems for a countable family of nonexpansive multivalued mappings. Demonstratio Mathematica, 54(1):47–67, 2021. https://doi.org/0.1515/dema-2021-0006
A.O.-E. Owolabi, T.O. Alakoya, A. Taiwo and O.T. Mewomo. A new inertialprojection algorithm for approximating common solution of variational inequality and fixed point problems of multivalued mappings. Numerical Algebra, Control & Optimization, 2021, 2021. https://doi.org/10.3934/naco.2021004
B.T. Polyak. Introduction to optimization. Translations series in mathematics and engineering. Optimization Software, Publications Division, New York, 1987.
P. Santos and S. Scheimberg. An inexact subgradient algorithm for equilibrium problem. Computational & Applied Mathematics, 30(1):91–107, 2011. https://doi.org/10.1590/S1807-03022011000100005
Y. Shehu and P. Cholamjiak. Iterative method with inertial for variational inequalities in Hilbert spaces. Calcolo, 56(4), 2019. https://doi.org/10.1007/s10092-018-0300-5
Y. Shehu, Q.L. Dong and D. Jiang. Single projection method for pseudomonotone variational inequality in Hilbert spaces. Optimization, 68(1):385–409, 2019. https://doi.org/10.1080/02331934.2018.1522636
S. Suantai, S. Kesornprom and P. Cholamjiak. Modified proximal algorithms for finding solutions of the split variational inclusions. Mathematics, 7(8), 2019. https://doi.org/10.3390/math7080708
S. Suantai, Y. Shehu, P. Cholamjiak and O.S. Iyiola. Strong convergence of a self-adaptive method for the split feasibility problem in Banach spaces. Journal of Fixed Point Theory and Applications, 20(68), 2018. https://doi.org/10.1007/s11784-018-0549-y
A. Taiwo, T.O. Alakoya and O.T. Mewomo. Halpern-type iterative process for solving split common fixed point and monotone variational inclusion problem between Banach spaces. Numerical Algorithms, 86(4):1359–1389, 2021. https://doi.org/10.1007/s11075-020-00937-2
A. Taiwo, T.O. Alakoya and O.T. Mewomo. Strong convergence theorem for fixed points of relatively nonexpansive multi-valued mappings and equilibrium problems in Banach spaces. Asian-European Journal of Mathematics, 14(08):2150137, 2021. https://doi.org/10.1142/S1793557121501370
L.H. Yen, L.D. Muu and N.T.T. Huyen. An algorithm for a class of split feasibility problems: application to a model in electricity production. Mathematical Methods of Operations Research, 84:549–565, 2016. https://doi.org/10.1007/s00186-016-0553-1
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. Journal of Industrial & Management Optimization, 18(1):239–265, 2022. https://doi.org/10.3934/jimo.2020152
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. Annali Dell’Universita’ Di Ferrara, 67(1):1–31, 2021. https://doi.org/10.1007/s11565-020-00354-2
P.N. Anh and L.D. Muu. A hybrid subgradient algorithm for nonexpansive mapping and equilibrium problems. Optimization Letters, 8:727–738, 2014.
E. Blum. From optimization and variational inequalities to equilibrium problems. The Mathematics Student, 63(1-4):123–145, 2014.
C.S. Chuang. Strong convergence theorems for the split variational inclusion problem in Hilbert spaces. Fixed Point Theory and Applications, 350(2013), 2013. https://doi.org/10.1186/1687-1812-2013-350
C.S. Chuang. Algorithms with new parameter conditions for split variational inclusion problems in Hilbert spaces with application to split feasibility problem. Optimization, 65:859–876, 2016. https://doi.org/10.1080/02331934.2015.1072715
K. Goebel and W.A. Wirk. Topics in metric fixed point theory. Cambridge studies in Advanced Mathematics, 28, Cambridge University Press, Cambridge, 1990.
H. Iiduka. Acceleration method for convex optimization over the fixed point set of a nonexpansive mappings. Mathematical Programming, 149:131–165, 2015. https://doi.org/10.1007/s10107-013-0741-1
C. Izuchukwu, A.A. Mebawondu and O.T. Mewomo. A new method for solving split variational inequality problems without co-coerciveness. Journal of Fixed Point Theory and Applications, 22(98), 2020. https://doi.org/10.1007/s11784-020-00834-0
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(2):387–412, 2021. https://doi.org/10.1080/02331934.2020.1716752
S. Kesornprom and P. Cholamjiak. Proximal type algorithms involving linesearch and inertial technique for split variational inclusion problem in Hilbert spaces with applications. Optimization, 68(12):2365–2391, 2019. https://doi.org/10.1080/02331934.2019.1638389
P.E. Maing´e. A hybrid extragradient-viscosity method for monotone operators and fixed point problems. SIAM Journal on Control and Optimization, 47:1499– 1515, 2008. https://doi.org/10.1137/060675319
A. Moudafi. Split monotone variational inclusions. Journal of Optimization Theory and Applications, 150:275–283, 2011. https://doi.org/10.1007/s10957-011-9814-6
A. Moudafi and B.S. Thakur. Solving proximal split feasibility problems without prior knowledge of operator norms. Optimization Letters, 8:2099–2110, 2014. https://doi.org/10.1007/s11590-013-0708-4
G.N. Ogwo, C. Izuchukwu and O.T. Mewomo. Inertial methods for finding minimum-norm solutions of the split variational inequality problem beyond monotonicity. Numerical Algorithms volume, 88:1419–1456, 2021. https://doi.org/10.1007/s11075-021-01081-1
M.A. Olona, T.O. Alakoya, A.O.-E. Owolabi and O.T. Mewomo. Inertial algorithm for solving equilibrium, variational inclusion and fixed point problems for an infinite family of strictly pseudocontractive mappings. Journal of Nonlinear Functional Analysis, 2021:1–21, 2021. https://doi.org/10.23952/jnfa.2021.10
M.A. Olona, T.O. Alakoya, A.O.-E. Owolabi and O.T. Mewomo. Inertial shrinking projection algorithm with self-adaptive step size for split generalized equilibrium and fixed point problems for a countable family of nonexpansive multivalued mappings. Demonstratio Mathematica, 54(1):47–67, 2021. https://doi.org/0.1515/dema-2021-0006
A.O.-E. Owolabi, T.O. Alakoya, A. Taiwo and O.T. Mewomo. A new inertialprojection algorithm for approximating common solution of variational inequality and fixed point problems of multivalued mappings. Numerical Algebra, Control & Optimization, 2021, 2021. https://doi.org/10.3934/naco.2021004
B.T. Polyak. Introduction to optimization. Translations series in mathematics and engineering. Optimization Software, Publications Division, New York, 1987.
P. Santos and S. Scheimberg. An inexact subgradient algorithm for equilibrium problem. Computational & Applied Mathematics, 30(1):91–107, 2011. https://doi.org/10.1590/S1807-03022011000100005
Y. Shehu and P. Cholamjiak. Iterative method with inertial for variational inequalities in Hilbert spaces. Calcolo, 56(4), 2019. https://doi.org/10.1007/s10092-018-0300-5
Y. Shehu, Q.L. Dong and D. Jiang. Single projection method for pseudomonotone variational inequality in Hilbert spaces. Optimization, 68(1):385–409, 2019. https://doi.org/10.1080/02331934.2018.1522636
S. Suantai, S. Kesornprom and P. Cholamjiak. Modified proximal algorithms for finding solutions of the split variational inclusions. Mathematics, 7(8), 2019. https://doi.org/10.3390/math7080708
S. Suantai, Y. Shehu, P. Cholamjiak and O.S. Iyiola. Strong convergence of a self-adaptive method for the split feasibility problem in Banach spaces. Journal of Fixed Point Theory and Applications, 20(68), 2018. https://doi.org/10.1007/s11784-018-0549-y
A. Taiwo, T.O. Alakoya and O.T. Mewomo. Halpern-type iterative process for solving split common fixed point and monotone variational inclusion problem between Banach spaces. Numerical Algorithms, 86(4):1359–1389, 2021. https://doi.org/10.1007/s11075-020-00937-2
A. Taiwo, T.O. Alakoya and O.T. Mewomo. Strong convergence theorem for fixed points of relatively nonexpansive multi-valued mappings and equilibrium problems in Banach spaces. Asian-European Journal of Mathematics, 14(08):2150137, 2021. https://doi.org/10.1142/S1793557121501370
L.H. Yen, L.D. Muu and N.T.T. Huyen. An algorithm for a class of split feasibility problems: application to a model in electricity production. Mathematical Methods of Operations Research, 84:549–565, 2016. https://doi.org/10.1007/s00186-016-0553-1