Abstract
Based on the very recent work by Censor-Gibali-Reich (http://arxiv.org/abs/1009.3780), we propose an extension of their new variational problem (Split Variational Inequality Problem) to monotone variational inclusions. Relying on the Krasnosel’skii-Mann Theorem for averaged operators, we analyze an algorithm for solving new split monotone inclusions under weaker conditions. Our weak convergence results improve and develop previously discussed Split Variational Inequality Problems, feasibility problems and related problems and algorithms.
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Censor, Y., Gibali, A., Reich, S.: The split variational inequality problem. The Technion-Israel Institute of Technology, Haifa (2010). arXiv:1009.3780
Censor, Y., Bortfeld, T., Martin, B., Trofimov, A.: A unified approach for inversion problems intensity-modulated radiation therapy. Phys. Med. Biol. 51, 2353–2365 (2006)
Byrne, C.: Iterative oblique projection onto convex sets and the split feasibility problem. Inverse Probl. 18, 441–453 (2002)
Combettes, P.L.: The convex feasibility problem in image recovery. Adv. Imaging Electron Phys. 95, 155–270 (1996)
Goldstein, E., Tretyakov, N.: Modified Lagrangians and Monotone Maps in Optimization. Wiley, New York (1996)
Byrne, C.: A unified treatment for some iterative algorithms in signal processing and image reconstruction. Inverse Probl. 20, 103–120 (2004)
Baillon, J.-B., Bruck, R.E., Reich, S.: On the asymptotic behavior of nonexpansive mappings and semigroups in Banach spaces. Houst. J. Math. 4, 1–9 (1978)
Aoyama, K., Kimura, Y., Takahashi, W.: Maximal monotone operators and maximal monotone functions for equilibrium problems. J. Convex Anal. 15(2), 395–409 (2008)
Rockafellar, R.T.: Monotone operators associated with saddle functions and minimax problems. Nonlinear analysis, Part I. In: Browder, F.E. (ed.) Symposia in Pure Mathematics, vol. 18, pp. 397–407. AMS, Providence (1970)
Masad, E., Reich, S.: A note on the multiple-set split feasibility problem in Hilbert space. J. Nonlinear Convex Anal. 8, 367–371 (2007)
Reich, S.: Averaged mappings in Hilbert ball. J. Math. Anal. Appl. 109, 199–206 (1985)
Lopez, G., Martin-Marquez, V., Xu, H.-K.: Iterative algorithms for the multi-sets feasibility problem. In: Censor, Y., Jiang, M., Wang, G. (eds.) Biomedical Mathematics: Promising Directions in Imaging, Therapy Planning and Inverse Problems, pp. 243–279. Medical Physics Publishing, Madison (2010)
Bruck, R.E., Reich, S.: Nonexpansive projections and resolvents of accretive operators in Banach spaces. Houst. J. Math. 3, 459–470 (1977)
Mann, W.: Mean value methods in iteration. Proc. Am. Math. Soc. 4, 506–510 (1953)
Moudafi, A.: The split common fixed-point problem for demicontractive mappings. Inverse Probl. 26, 1–6 (2010)
Xu, H.K.: A variable Krasnosel’skii-Mann algorithm and the multiple-set split feasibility problem. Inverse Probl. 22, 2021–2034 (2006)
Censor, Y., Segal, A.: The split common fixed point problem for directed operators. J. Convex Anal. 16, 587–600 (2009)
Bauschke, H.H., Combettes, P.: A weak-to-strong convergence principle for Fejér monotone methods in Hilbert spaces. Math. Methods Oper. Res. 26, 248–264 (2001)
Bauschke, H.H., Borwein, J.M.: On projection algorithms for solving convex feasibility problems. SIAM Rev. 38(3), 367–426 (1996)
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Communicated by M. Théra.
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Moudafi, A. Split Monotone Variational Inclusions. J Optim Theory Appl 150, 275–283 (2011). https://doi.org/10.1007/s10957-011-9814-6
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DOI: https://doi.org/10.1007/s10957-011-9814-6