Abstract
We introduce a new optimization approach to solving systems of split equality problems in real Hilbert spaces. We use the inertial method in order to improve the convergence rate of the proposed algorithms. Our algorithms do not depend on the norms of the bounded linear operators which appear in each split equality problem of the system under consideration. This is also a strong point of our algorithms because it is known that it is difficult to compute or estimate the norm of a linear operator in the general case.
Similar content being viewed by others
Availability of data and materials
Not applicable.
References
Alvarez, F., Attouch, H.: An inertial proximal method for maximal monotone operators via discretization of a nonlinear oscillator with damping. Set-Valued Anal. 9, 3–11 (2001)
Alvarez, F.: Weak convergence of a relaxed and inertial hybrid projectionproximal point algorithm for maximal monotone operators in Hilbert space. SIAM J. Optim. 14(3), 773–782 (2004)
Attouch, H., Bolte, J., Redont, P., Soubeyran, A.: A new class of alternating proximal minimization algorithms with costs-to-move. SIAM J. Optim. 18, 1061–1081 (2007)
Attouch, H., Bolte, J., Redont, P., Soubeyran, A.: Alternating proximal algorithms for weakly coupled minimization problems. Applications to dynamical games and PDE’s. J. Convex Anal. 15, 485–506 (2008)
Attouch, H., Cabot, A.: Convergence rates of inertial forward-backward algorithms. SIAM J. Optim. 28, 849–874 (2018)
Attouch, H., Cabot, A.: Convergence of damped inertial dynamics governed by regularized maximally monotone operators. J. Differ. Equ. 264, 7138–7182 (2018)
Attouch, H., Cabot, A.: Convergence of a relaxed inertial proximal algorithm for maximally monotone operators. Math. Program. 184, 243–287 (2020)
Attouch, H., László, S.C.: Continuous Newton-like inertial dynamics for monotone inclusions. Set-Valued Var. Anal. 29, 555–581 (2021)
Attouch, H., Peypouquet, J.: Convergence of inertial dynamics and proximal algorithms governed by maximal monotone operators. Math. Program. Ser. B. 174, 391–432 (2019)
Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer, Berlin (2011)
Beck, A., Teboulle, M.: A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM J. Imaging Sci. 2(1), 183–202 (2009)
Censor, Y., Elfving, T.: A multiprojection algorithm using Bregman projections in a product space. Numer. Algorithms 8, 221–239 (1994)
Censor, Y., Bortfeld, T., Martin, B., Trofimov, A.: A unified approach for inversion problems in intensity-modulated radiation therapy. Phys. Med. Biol. 51, 2353–2365 (2006)
Combettes, P.L., Glaudin, L.E.: Quasinonexpansive iterations on the affine hull of orbits: from Mann’s mean value algorithm to inertial methods. SIAM J. Optim. 27, 2356–2380 (2017)
Chang, S.-S., Yang, L., Qin, L., Ma, Z.: Strongly convergent iterative methods for split equality variational inclusion problems in banach spaces. Acta Math. Sci. 36, 1641–1650 (2016)
Eslamian, M., Shehu, Y., Iyiola, O.S.: A strong convergence theorem for a general split equality problem with applications to optimization and equilibrium problem. Calcolo 55(48), 1–31 (2018)
Goebel, K., Reich, S.: Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, Marcek Dekker (1984)
Izuchukwu, C., Mewomo, O.T., Okeke, C.C.: Systems of variational inequalities and multiple-set split equality fixed-point problems for countable families of multivalued type-one mappings of the demicontractive type. Ukr. Math. J. 71(11), 1692–1718 (2020)
Kazmi, K.R., Ali, R., Furkan, M.: Common solution to a split equality monotone variational inclusion problem, a split equality generalized general variational-like inequality problem and a split equality fixed point problem. Fixed Point Theory 20(1), 211–232 (2019)
Kopecká, E., Reich, S.: A note on alternating projections in Hilbert space. J. Fixed Point Theory appl. 12, 41–47 (2012)
Moudafi, A., Al-Shemas, E.: Simultaneous Iterative Methods for split equality problems. Trans. Math. Program. Appl. 1(2), 1–11 (2013)
Moudafi, A.: Alternating CQ-algorithms for convex feasibility and split fixed-point problems. J. Nonlinear Convex Anal. 15, 809–818 (2014)
Opial, Z.: Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bull. Am. Math. Soc. 73(4), 591–597 (1967)
Polyak, B.T.: Some methods of speeding up the convergence of iteration methods. USSR Comput. Math. Math. Phys. 4(5), 1–17 (1964)
Reich, S., Tuyen, T.M.: A new approach to solving split equality problems in Hilbert spaces. Optimization 71(15), 4423–4445 (2022)
Reich, S., Tuyen, T.M., Ha, M.T.N.: The split feasibility problem with multiple output sets in Hilbert spaces. Optim. Lett. 14, 2335–2353 (2020)
Reich, S., Tuyen, T.M., Ha, M.T.N.: An optimization approach to solving the split feasibility problem in Hilbert spaces. J. Global Optim. 79, 837–852 (2021)
Reich, S., Tuyen, T.M., Ha, M.T.N.: A product space approach to solving the split common fixed point problem in Hilbert spaces. J. Nonlinear Convex Anal. 21(11), 2571–2588 (2021)
Saejung, S., Yotkaew, P.: Approximation of zeros of inverse strongly monotone operators in Banach spaces. Nonlinear Anal. 75, 742–750 (2012)
Sahu, D.R., Cho, Y.J., Dong, Q.-L., Kashyap, M., Li, X.: Inertial relaxed CQ algorithms for solving a split feasibility problem in Hilbert spaces. Numer. Algorithm 87, 1075–1095 (2021)
Taiwo, A., Jolaoso, L.O., Mewomo, O.T.: A modified Halpern algorithm for approximating a common solution of split equality convex minimization problem and fixed point problem in uniformly convex Banach spaces. Comp. Appl. Math. 38(77), 1–28 (2019)
Taiwo, A., Jolaoso, L.O., Mewomo, O.T.: General alternative regularization method for solving split equality common fixed point problem for quasi-pseudocontractive mappings in Hilbert spaces. Ricerche mat. 69, 235–259 (2020)
Taiwo, A., Jolaoso, L.O., Mewomo, O.T.: Viscosity approximation method for solving the multiple-set split equality common fixed-point problems for quasi-pseudocontractive mappings in Hilbert spaces. J. Ind. Manag. Optim. 17(5), 2733–2759 (2021)
Tuyen, T.M.: Regularization methods for the split equality problems in Hilbert spaces. Bull. Malays. Math. Sci. Soc. 46(1), 44 (2023)
Vuong, P.T., Strodiot, J.J., Nguyen, V.H.: A gradient projection method for solving split equality and split feasibility problems in Hilbert spaces. Optimization 64, 2321–2341 (2015)
Yang, Q.: The relaxed CQ algorithm solving the split feasibility problem. Inverse Prob. 20(4), 1261 (2004)
Zhao, J.: Solving split equality fixed-point problem of quasi-nonexpansive mappings without prior knowledge of operators norms. Optimization 64(15), 2619–2630 (2015)
Acknowledgements
All the authors are grateful to the editors and to an anonymous referee for their useful comments and helpful suggestions.
Funding
Simeon Reich was partially supported by the Israel Science Foundation (Grant 820/17), by the Fund for the Promotion of Research at the Technion (Grant 2001893) and by the Technion General Research Fund (Grant 2016723). Truong Minh Tuyen and Nguyen Song Ha were supported by the Science and Technology Fund of the Thai Nguyen University of Sciences.
Author information
Authors and Affiliations
Contributions
All authors wrote the main manuscript text and reviewed the manuscript.
Corresponding authors
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Ethical Approval
Not applicable.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Reich, S., Tuyen, T.M. & Ha, N.S. A new optimization approach to solving split equality problems in Hilbert spaces. J Glob Optim (2024). https://doi.org/10.1007/s10898-024-01389-x
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s10898-024-01389-x