Computing the Resolvent of the Sum of Maximally Monotone Operators with the Averaged Alternating Modified Reflections Algorithm
- 229 Downloads
The averaged alternating modified reflections algorithm is a projection method for finding the closest point in the intersection of closed and convex sets to a given point in a Hilbert space. In this work, we generalize the scheme so that it can be used to compute the resolvent of the sum of two maximally monotone operators. This gives rise to a new splitting method, which is proved to be strongly convergent. A standard product space reformulation permits to apply the method for computing the resolvent of a finite sum of maximally monotone operators. Based on this, we propose two variants of such parallel splitting method.
KeywordsMaximally monotone operator Resolvent Averaged alternating modified reflections algorithm Douglas–Rachford algorithm Splitting method
Mathematics Subject Classification47H05 47J25 65K05 47N10
We greatly appreciate the constructive comments of two anonymous reviewers which helped us to improve the paper. This work was partially supported by Ministerio de Economía, Industria y Competitividad (MINECO) of Spain and European Regional Development Fund (ERDF), grant MTM2014-59179-C2-1-P. FJAA was supported by the Ramón y Cajal program by MINECO and ERDF (RYC-2013-13327) and RC was supported by MINECO and European Social Fund (BES-2015-073360) under the program “Ayudas para contratos predoctorales para la formación de doctores 2015”.
- 2.Aragón Artacho, F.J., Campoy, R.: Optimal rates of linear convergence of the averaged alternating modified reflections method for two subspaces. Numer. Algor. 1–25 (2018). https://doi.org/10.1007/s11075-018-0608-x
- 3.Bauschke, H.H.; Burachik, R.S., Kaya, C.Y.: Constraint splitting and projection methods for optimal control of double integrator. ArXiv e-prints: arXiv:1804.03767 (2018)
- 10.Adly, S., Bourdin, L., Caubet, F.: On a decomposition formula for the proximal operator of the sum of two convex functions. J. Convex Anal. 26(3) (2019). http://www.heldermann.de/JCA/JCA26/JCA263/jca26037.htm