We introduce a new general proximal-point algorithm for an infinite family of monotone operators in a real Hilbert space and establish strong convergence of the iterative process to a common null point of the infinite family of monotone operators. Our result generalizes and improves numerous results in the available literature.
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References
H. Brezis, Operateurs Maximaux Monotones et Semi-Groups de Contractions Dans les Espaces de Hilbert, North-Holland, Amsterdam (1973).
G. Morosanu, Nonlinear Evolution Equations and Applications, Reidel (1988).
P. M. Pardalos, T. M. Rassias, and A. A. Khan, Nonlinear Analysis and Variational Problems, Springer, Berlin (2010).
R. S. Burachik and A. N. Iusem, Set-Valued Mappings and Enlargements of Monotone Operators, Springer, New York (2008).
R. T. Rockafellar, “Monotone operators and the proximal point algorithm,” SIAM J. Control Optim., 14, 877–898 (1976).
O. Güler, “On the convergence of the proximal point algorithm for convex minimization,” SIAM J. Control Optim., 29, 403–419 (1991).
M. V. Solodov and B. F. Svaiter, “Forcing strong convergence of proximal point iterations in a Hilbert space,” Math. Program., Ser. A, 87, 189–202 (2000).
S. Kamimura and W. Takahashi, “Approximating solutions of maximal monotone operators in Hilbert spaces,” J. Approx. Theory, 106, 226–240 (2000).
S. Kamimura and W. Takahashi, “Strong convergence of a proximal-type algorithm in a Banach space,” SIAM J. Optim., 13, 938–945 (2002).
W. Takahashi, “Viscosity approximation methods for resolvents of accretive operators in Banach spaces,” J. Fixed Point Theory Appl., 1, 135–147 (2007).
H. K. Xu, “Iterative algorithms for nonlinear operators,” J. London Math. Soc., 66, 240–256 (2002).
O. A. Boikanyo and G. Morosanu, “Modified Rockafellars algorithms,” Math. Sci. Res. J., 12, 101–122 (2009).
H. K. Xu, “A regularization method for the proximal point algorithm,” J. Global. Optim., 36, 115–125 (2006).
N. Lehdili and A. Moudafi, “Combining the proximal algorithm and Tikhonov method,” Optimization, 37, 239–252 (1996).
O. A. Boikanyo and G. Morosanu, “A proximal point algorithm converging strongly for general errors,” Optim. Lett., 4, 635–641 (2010).
O. A. Boikanyo and G. Morosanu, “Inexact Halpern-type proximal point algorithm,” J. Global Optim., 51, 11–26 (2011).
C. A. Tian and Y. Song, “Strong convergence of a regularization method for Rockafellars proximal point algorithm,” J. Global Optim., DOI 10.1007/s10898-011-9827-6.
Y. Yao and N. Shahzad, “Strong convergence of a proximal point algorithm with general errors,” Optim. Lett., DOI 10.1007/s11590-011-0286-2.
Y. Song and C. Yang, “A note on a paper A regularization method for the proximal point algorithm,” J. Global Optim., 43, 171–174 (2009).
F. Wang, “A note on the regularized proximal point algorithm,” J. Global Optim., 50, 531–535 (2011).
X. Chai, B. Li, and Y. Song, “Strong and weak convergence of the modified proximal point algorithms in Hilbert space,” Fixed Point Theory Appl., 2010, Article ID 240450, 11 p. (2010).
F. Wang and C. Huanhuan, “On the contraction-proximal point algorithms with multi-parameters,” J. Global Optim., DOI 10.1007/s10898-011-9772-4.
Y. Yao and M. A. Noor, “On convergence criteria of generalized proximal point algorithms,” J. Comput. Appl. Math., 217, 46–55 (2008).
W. Takahashi, Nonlinear Functional Analysis, Fixed Point Theory, and Its Applications, Yokohama Publ., Yokohama (2000).
G. Marino and H. K. Xu, “A general iterative method for nonexpansive mappings in Hilbert spaces,” J. Math. Anal. Appl., 318, 43–52 (2006).
A. Moudafi, “Viscosity approximation methods for fixed-point problems,” J. Math. Anal. Appl., 241, 46–55 (2000).
S. S. Chang, J. K. Kim, and X. R. Wang, “Modified block iterative algorithm for solving convex feasibility problems in Banach spaces,” J. Inequal. Appl., 2010, No. 14, Article ID 869684 (2010).
P. E. Mainge, “Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization,” Set-Valued Analysis, 16, 899–912 (2008).
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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 68, No. 11, pp. 1483–1492, November, 2016.
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Eslamian, M., Vahidi, J. General Proximal-Point Algorithm for Monotone Operators. Ukr Math J 68, 1715–1726 (2017). https://doi.org/10.1007/s11253-017-1322-x
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DOI: https://doi.org/10.1007/s11253-017-1322-x