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Inexact Halpern-type proximal point algorithm. (English) Zbl 1295.47073

Summary: We present several strong convergence results for the modified, Halpern-type, proximal point algorithm \({x_{n+1}=\alpha_{n}u+(1-\alpha_{n})J_{\beta_n}x_n+e_{n}}\) (\(n = 0, 1, \dots\); \({u,x_0\in H}\) given, and \({J_{\beta_n}=(I+\beta_nA)^{-1}}\), for a maximal monotone operator \(A\)) in a real Hilbert space, under new sets of conditions on \({\alpha_n\in(0,1)}\) and \({\beta_n\in(0,\infty)}\). These conditions are weaker than those known to us and our results extend and improve some recent results such as those of H. K. Xu [J. Lond. Math. Soc., II. Ser. 66, No. 1, 240–256 (2002; Zbl 1013.47032); J. Glob. Optim. 36, No. 1, 115–125 (2006; Zbl 1131.90062)]. We also show how to apply our results to approximate minimizers of convex functionals. In addition, we give convergence rate estimates for a sequence approximating the minimum value of such a functional.

MSC:

47J25 Iterative procedures involving nonlinear operators
47H05 Monotone operators and generalizations
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
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