Bauschke and Moursi have recently obtained results that implicitly contain the fact that the composition of finitely many averaged mappings on a Hilbert space that have approximate fixed points also has approximate fixed points and thus is asymptotically regular. Using techniques of proof mining, we analyze their arguments to obtain effective uniform rates of asymptotic regularity.
This is a preview of subscription content, access via your institution.
Buy single article
Instant access to the full article PDF.
Price includes VAT (USA)
Tax calculation will be finalised during checkout.
Bauschke, H.: The composition of projections onto closed convex sets in Hilbert space is asymptotically regular. Proc. Am. Math. Soc. 131(1), 141–146 (2003)
Bauschke, H., Borwein, J., Lewis, A.: The method of cyclic projections for closed convex sets in Hilbert space. In: Censor, Y., Reich, S. (eds.) Recent Developments in Optimization Theory and Nonlinear Analysis (Jerusalem, 1995), Contemporary Mathematics 204, pp. 1–38. American Mathematical Society, Providence (1997)
Bauschke, H., Combettes, P.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces, 2nd edn. Springer, Berlin (2017)
Bauschke, H., Martín-Márquez, V., Moffat, S., Wang, X.: Compositions and convex combinations of asymptotically regular firmly nonexpansive mappings are also asymptotically regular. Fixed Point Theory Appl. 2012, 53 (2012)
Bauschke, H., Moursi, W.: The magnitude of the minimal displacement vector for compositions and convex combinations of firmly nonexpansive mappings. Optim. Lett. 12(7), 1465–1474 (2018)
Bauschke, H., Moursi, W.: On the minimal displacement vector of compositions and convex combinations of nonexpansive mappings. Found. Comput. Math. 20, 1653–1666 (2020)
Bregman, L.M.: The method of successive projection for finding a common point of convex sets. Soviet Math. Dokl. 6, 688–692 (1965)
Brézis, H., Haraux, A.: Image d’une somme d’opérateurs monotones et applications. Isr. J. Math. 23(2), 165–186 (1976)
Browder, F.E., Petryshyn, W.V.: The solution by iteration of nonlinear functional equations in Banach spaces. Bull. Am. Math. Soc. 72, 571–575 (1966)
Bruck, R.E., Reich, S.: Nonexpansive projections and resolvents of accretive operators in Banach spaces. Houst. J. Math. 3(4), 459–470 (1977)
Combettes, P.: Solving monotone inclusions via compositions of nonexpansive averaged operators. Optimization 53(5–6), 475–504 (2004)
Giselsson, P.: Tight global linear convergence rate bounds for Douglas–Rachford splitting. J. Fixed Point Theory Appl. 19(4), 2241–2270 (2017)
Kohlenbach, U.: Applied Proof Theory: Proof Interpretations and Their Use in Mathematics. Springer Monographs in Mathematics. Springer, Berlin (2008)
Kohlenbach, U.: On the quantitative asymptotic behavior of strongly nonexpansive mappings in Banach and geodesic spaces. Isr. J. Math. 216(1), 215–246 (2016)
Kohlenbach, U.: A polynomial rate of asymptotic regularity for compositions of projections in Hilbert space. Found. Comput. Math. 19(1), 83–99 (2019)
Kohlenbach, U.: Proof-theoretic methods in nonlinear analysis. In: Sirakov, B., Ney de Souza, P., Viana, M. (eds.) Proceedings of the International Congress of Mathematicians 2018 (ICM 2018), vol. 2, pp. 61–82. World Scientific, Singapore (2019)
Moursi, W., Vandenberghe, L.: Douglas–Rachford splitting for the sum of a Lipschitz continuous and a strongly monotone operator. J. Optim. Theory Appl. 183(1), 179–198 (2019)
I would like to thank Ulrich Kohlenbach for pointing me to the paper  and for suggesting an improvement of Proposition 2.7. This work has been supported by the German Science Foundation (DFG Project KO 1737/6-1) and by a grant of the Romanian Ministry of Research, Innovation and Digitization, CNCS/CCCDI – UEFISCDI, Project Number PN-III-P1-1.1-PD-2019-0396, within PNCDI III.
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
About this article
Cite this article
Sipoş, A. Quantitative inconsistent feasibility for averaged mappings. Optim Lett (2021). https://doi.org/10.1007/s11590-021-01812-2
- Proof mining
- Averaged mappings
- Nonexpansive mappings
- Rates of asymptotic regularity