Abstract
This note is devoted to the splitting algorithm proposed by Davis and Yin (Set-valued Var. Anal. 25(4), 829–858, 2017) for computing a zero of the sum of three maximally monotone operators, with one of them being cocoercive. We provide a direct proof that guarantees its convergence when the stepsizes are smaller than four times the cocoercivity constant, thus doubling the size of the interval established by Davis and Yin. As a by-product, the same conclusion applies to the forward-backward splitting algorithm. Further, we use the notion of “strengthening” of a set-valued operator to derive a new splitting algorithm for computing the resolvent of the sum. Last but not least, we provide some numerical experiments illustrating the importance of appropriately choosing the stepsize and relaxation parameters of the algorithms.
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References
Aragón Artacho, F.J., Campoy, R., Tam, M.K.: Strengthened splitting methods for computing resolvents. Comput. Optim. Appl. 80, 549–585 (2021)
Aragón Artacho, F.J., Censor, Y., Gibali, A.: The cyclic Douglas–Rachford algorithm with r-sets-Douglas–Rachford operators. Optim. Methods Softw. 34(4), 875–889 (2019)
Attouch, H., Peypouquet, J., Redont, P.: Backward-forward algorithms for structured monotone inclusions in Hilbert spaces. J. Math. Anal. Appl. 457, 1095–1117 (2018)
Baillon, J. -B., Haddad, G.: Quelques propriétés des opérateurs angle-bornés et n-cycliquement monotones. Israel. J. Math. 26, 137–150 (1997)
Bartz, S., Dao, M.N., Phan, H.M.: Conical averagedness and convergence analysis of fixed point algorithms. J. Glob Optim. https://doi.org/10.1007/s10898-021-01057-4(2021)
Beck, A., Teboulle, M.: A Fast Iterative Shrinkage-Tresholding Algorithm for linear inverse problems. SIAM J. Imaging Sci. 2(1), 183–202 (2009)
Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces, 2nd edn. Springer, Berlin (2017)
Borwein, J.M., Tam, M.T.: A cyclic Douglas–Rachford iteration scheme. J. Optim. Theory Appl. 160, 1–29 (2014)
Campoy, R.: A product space reformulation with reduced dimension for splitting algorithms. arXiv:1910.14185 (2021)
Cegielski, A.: Iterative Methods for Fixed Point Problems in Hilbert Spaces Lecture Notes in Mathematics, vol. 2057. Springer, Heidelberg (2012)
Combettes, P.L., Pesquet, J.-C.: Proximal Splitting Methods in Signal Processing. In: Bauschke, H.H., Burachik, R.S., Combettes, P.L., Elser, V., Luke, D.R., Wolkowicz, H (eds.) Fixed-Point Algorithms for Inverse Problems in Science and Engineering, pp 185–212. Springer, New York (2011)
Combettes, P.L., Yamada, I.: Compositions and convex combinations of averaged nonexpansive operators. J. Math. Anal. Appl. 425, 55–70 (2015)
Condat, L., Kitahara, D., Contreras, A., Hirabayashi, A.: Proximal splitting algorithms for convex optimization: A tour of recent advances, with new twists. arXiv:1912.00137 (2021)
Dao, M.N., Dizon, N., Hogan, J.A., Tam, M.K.: Constraint reduction reformulations for projection algorithms with applications to wavelet construction. J. Optim. Theory Appl. 190, 201–233 (2021)
Dao, M.N., Phan, H.M.: Adaptive Douglas–Rachford splitting algorithm for the sum of two operators. SIAM J. Optim. 29(4), 2697–2724 (2019)
Dao, M.N., Phan, H.M.: Computing the resolvent of the sum of operators with application to best approximation problems. Optim. Lett. 14, 1193–1205 (2020)
Dao, M.N., Phan, H.M.: An adaptive splitting algorithm for the sum of two generalized monotone operators and one cocoercive operator. Fixed Point Theory Algorithms Sci. Eng. 2021, 16 (2021). https://doi.org/10.1186/s13663-021-00701-8
Davis, D., Yin, W.: A three-operator splitting scheme and its optimization applications. Set-valued Var. Anal. 25(4), 829–858 (2017)
Giselsson, P.: Nonlinear forward-backward splitting with projection correction. SIAM J. Optim. 31(3), 2199–2226 (2021)
Giselsson, P., Moursi, W.M.: On compositions of special cases of Lipschitz continuous operators. Fixed Point Theory Algorithms Sci. Eng. 2021, 25 (2021). https://doi.org/10.1186/s13663-021-00709-0
Latafat, P., Patrinos, P.: Asymmetric forward-backward-adjoint splitting for solving monotone inclusions involving three operators. Comput. Optim. Appl. 68, 57–93 (2017)
Lions, P.L., Mercier, B.: Splitting algorithms for the sum of two nonlinear operators. SIAM J. Numer. Anal. 16(6), 964–979 (1979)
Malitsky, Y., Tam, M.K.: Resolvent splitting for sums of monotone operators with minimal lifting. arXiv:2108.02897 (2021)
Minty, G.: Monotone (nonlinear) operators in a Hilbert space. Duke Math. J. 29, 341–34 (1962)
Passty, G.B.: Ergodic convergence to a zero of the sum of monotone operators in Hilbert space. J. Math. Anal. Appl. 72, 383–390 (1979)
Pierra, G.: Decomposition through formalization in a product space. Math. Program. 28, 96–115 (1984)
Rieger, J., Tam, M.K.: Backward-forward-reflected-backward splitting for three operator monotone inclusions. Appl. Math. Comput. 381, 125248 (2020)
Ryu, E.K.: Uniqueness of DRS as the 2 operator resolvent-splitting and impossibility of 3 operator resolvent-splitting. Math. Program. 182, 233–273 (2020)
Ryu, E.K., Vũ, B.C.: Finding the forward-Douglas–Rachford-forward method. J. Optim. Theory Appl. 184, 858–876 (2020)
Acknowledgements
The authors would like to thank Patrick Combettes for making us aware of [17] right before submitting this work. We thank the referees for their careful reading and their constructive comments which helped improve our manuscript.
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FJAA and DTB were partially supported by the Ministry of Science, Innovation and Universities of Spain and the European Regional Development Fund (ERDF) of the European Commission, Grant PGC2018-097960-B-C22. FJAA was partially supported by the Generalitat Valenciana (AICO/2021/165). DTB was supported by MINECO and European Social Fund (PRE2019-090751) under the program “Ayudas para contratos predoctorales para la formación de doctores” 2019.
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Aragón-Artacho, F.J., Torregrosa-Belén, D. A Direct Proof of Convergence of Davis–Yin Splitting Algorithm Allowing Larger Stepsizes. Set-Valued Var. Anal 30, 1011–1029 (2022). https://doi.org/10.1007/s11228-022-00631-6
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DOI: https://doi.org/10.1007/s11228-022-00631-6