A self-adaptive step size algorithm for solving the split feasibility problem with multiple output sets in Hilbert spaces
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Abstract
The purpose of this manuscript is to introduce a new self- adaptive algorithm for solving the split feasibility problem with multiple output sets in real Hilbert spaces. Our algorithm leverages information from previous steps to guide its execution, thereby removing the need to compute or estimate the norm of the given operator. Lastly, we present a simple numerical example to illustrate the performance of our proposed algorithm.
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References
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[2] Anh T.V., An extragradient method for finding minimum-norm solution of the split equilibrium problem. Acta Math. Vietnam, 42, 587-604 (2017).
[3] Anh T.V., A parallel method for variational inequalities with the multiple-sets split feasibility problem constraints. J. Fixed Point Theory Appl. 19, 2681-2696 (2017).
[4] Anh T.V., Linesearch methods for bilevel split pseudomonotone variational inequality problems. Numer. Algorithms 81, 1067-1087 (2019).
[5] Anh T.V. and Muu L.D., A projection-fixed point method for a class of bilevel variational inequalities with split fixed point constraints. Optimization 65, 1229-1243 (2016).
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[7] Byrne C., Iterative oblique projection onto convex sets and the split feasibility problem. Inverse Probl. 18, 441-453 (2002).
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[12] Censor Y. and Elfving T., A multiprojection algorithm using Bregman
projections in a product space. Numer. Algorithms 8, 221-239 (1994).
[13] Censor Y., Elfving T., Kopf N. and Bortfeld T., The multiple-sets split feasibility problem and its applications for inverse problems. Inverse
Prob. 21, 2071-2084 (2005).
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[15] Eslamian M., A hierarchical variational inequality problem for general- ized demimetric mappings with applications. J. Nonlinear Var. Anal. 5, 965-979 (2021).
[16] Hai N.M., Van L.H.M. and Anh T.V., An Algorithm for a Class of Bilevel Variational Inequalities with Split Variational Inequality and Fixed Point Problem Constraints. Acta Math. Vietnam, 46, 515–530 (2021).
[17] Huy P.V., Hien N.D. and Anh T.V., A strongly convergent modified Halpern subgradient extragradient method for solving the split variational inequality problem. Vietnam J. Math. 48, 187-204 (2020).
[18] Huy P.V., Van L.H.M., Hien N.D. and Anh T.V., Modified Tseng’s extragradient methods with self-adaptive step size for solving bilevel split variational inequality problems. Optimization 71, 1721-1748 (2022).
[19] Iiduka H., Fixed point optimization algorithm and its application to power control in CDMA data networks. Math. Program. 133, 227-242 (2012).
[20] Iiduka H., Fixed point optimization algorithm and its application to network bandwidth allocation. J. Comput. Appl. Math. 236, 1733-1742 (2012).
[21] Konnov I.V., Combined Relaxation Methods for Variational Inequalities. Springer, Berlin (2000).
[22] Liu B., Qu B. and Zheng N., A Successive Projection Algorithm for Solving the Multiple-Sets Split Feasibility Problem. Numer. Funct. Anal. Optim. 35, 1459-1466 (2014).
[23] Maing ́e P.E., A hybrid extragradient-viscosity method for monotone op- erators and fixed point problems. SIAM J. Control Optim. 47, 1499-1515 (2008).
[24] Reich S., Truong M.T. and Mai T.N.H., The split feasibility problem with multiple output sets in Hilbert spaces. Optim. Lett. 14, 2335–2353 (2020).
[25] Wen M., Peng J.G. and Tang Y.C., A cyclic and simultaneous it- erative method for solving the multiple-sets split feasibility problem. J. Optim. Theory Appl. 166(3), 844-860 (2015).
[26] Yamada I. and Ogura N., Hybrid steepest descent method for vari- ational inequality problem over the fixed point set of certain quasi- nonexpansive mappings. Numer. Funct. Anal. Optim. 25, 619-655 (2004).
[27] Zhao J.L. and Yang Q.Z., A simple projection method for solving the multiple-sets split feasibility problem. Inverse Probl. Sci. Eng. 21, 537-546 (2013).
[28] Zhao J.L., Zhang Y.J. and Yang Q.Z., Modified projection meth- ods for the split feasibility problem and the multiple-sets split feasibility problem. Appl. Math. Comput. 219, 1644-1653 (2012).
[2] Anh T.V., An extragradient method for finding minimum-norm solution of the split equilibrium problem. Acta Math. Vietnam, 42, 587-604 (2017).
[3] Anh T.V., A parallel method for variational inequalities with the multiple-sets split feasibility problem constraints. J. Fixed Point Theory Appl. 19, 2681-2696 (2017).
[4] Anh T.V., Linesearch methods for bilevel split pseudomonotone variational inequality problems. Numer. Algorithms 81, 1067-1087 (2019).
[5] Anh T.V. and Muu L.D., A projection-fixed point method for a class of bilevel variational inequalities with split fixed point constraints. Optimization 65, 1229-1243 (2016).
[6] Buong N., Iterative algorithms for the multiple-sets split feasibility prob-
lem in Hilbert spaces. Numer. Algorithms 76, 783-798 (2017).
[7] Byrne C., Iterative oblique projection onto convex sets and the split feasibility problem. Inverse Probl. 18, 441-453 (2002).
[8] Byrne C., Censor Y., Gibali A. and Reich S., The split common null point problem. J. Nonlinear Convex Anal. 13, 759-775 (2012).
[9] Ceng L.C., Ansari Q.H. and Yao J.C., Relaxed extragradient meth- ods for finding minimum-norm solutions of the split feasibility problem. Nonlinear Anal. 75, 2116-2125 (2012).
[10] Ceng L.C., Petru ̧sel A., Qin X. and Yao J.C., Two inertial subgradient extragradient algorithms for variational inequalities with fixed-point constraints. Optimization 70, 1337-1358 (2021).
[11] Censor Y., Bortfeld T., Martin B. and Trofimov A., A unified approach for inversion problems in intensity-modulated radiation therapy. Phys. Med. Biol. 51, 2353-2365 (2006).
[12] Censor Y. and Elfving T., A multiprojection algorithm using Bregman
projections in a product space. Numer. Algorithms 8, 221-239 (1994).
[13] Censor Y., Elfving T., Kopf N. and Bortfeld T., The multiple-sets split feasibility problem and its applications for inverse problems. Inverse
Prob. 21, 2071-2084 (2005).
[14] Censor Y. and Segal A., Iterative projection methods in biomedical in- verse problems, in: Y. Censor, M. Jiang, A.K. Louis (Eds.), Mathematical Methods in Biomedical Imaging and Intensity-Modulated Therapy, IMRT, Edizioni della Norale, Pisa, Italy, 2008, pp. 65-96.
[15] Eslamian M., A hierarchical variational inequality problem for general- ized demimetric mappings with applications. J. Nonlinear Var. Anal. 5, 965-979 (2021).
[16] Hai N.M., Van L.H.M. and Anh T.V., An Algorithm for a Class of Bilevel Variational Inequalities with Split Variational Inequality and Fixed Point Problem Constraints. Acta Math. Vietnam, 46, 515–530 (2021).
[17] Huy P.V., Hien N.D. and Anh T.V., A strongly convergent modified Halpern subgradient extragradient method for solving the split variational inequality problem. Vietnam J. Math. 48, 187-204 (2020).
[18] Huy P.V., Van L.H.M., Hien N.D. and Anh T.V., Modified Tseng’s extragradient methods with self-adaptive step size for solving bilevel split variational inequality problems. Optimization 71, 1721-1748 (2022).
[19] Iiduka H., Fixed point optimization algorithm and its application to power control in CDMA data networks. Math. Program. 133, 227-242 (2012).
[20] Iiduka H., Fixed point optimization algorithm and its application to network bandwidth allocation. J. Comput. Appl. Math. 236, 1733-1742 (2012).
[21] Konnov I.V., Combined Relaxation Methods for Variational Inequalities. Springer, Berlin (2000).
[22] Liu B., Qu B. and Zheng N., A Successive Projection Algorithm for Solving the Multiple-Sets Split Feasibility Problem. Numer. Funct. Anal. Optim. 35, 1459-1466 (2014).
[23] Maing ́e P.E., A hybrid extragradient-viscosity method for monotone op- erators and fixed point problems. SIAM J. Control Optim. 47, 1499-1515 (2008).
[24] Reich S., Truong M.T. and Mai T.N.H., The split feasibility problem with multiple output sets in Hilbert spaces. Optim. Lett. 14, 2335–2353 (2020).
[25] Wen M., Peng J.G. and Tang Y.C., A cyclic and simultaneous it- erative method for solving the multiple-sets split feasibility problem. J. Optim. Theory Appl. 166(3), 844-860 (2015).
[26] Yamada I. and Ogura N., Hybrid steepest descent method for vari- ational inequality problem over the fixed point set of certain quasi- nonexpansive mappings. Numer. Funct. Anal. Optim. 25, 619-655 (2004).
[27] Zhao J.L. and Yang Q.Z., A simple projection method for solving the multiple-sets split feasibility problem. Inverse Probl. Sci. Eng. 21, 537-546 (2013).
[28] Zhao J.L., Zhang Y.J. and Yang Q.Z., Modified projection meth- ods for the split feasibility problem and the multiple-sets split feasibility problem. Appl. Math. Comput. 219, 1644-1653 (2012).