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 nxn) + e(n) (n = 0, 1,...;u, x(0) is an element of H given, and J(beta ...
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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 nxn) + e(n) (n = 0, 1,...;u, x(0) is an element of H given, and J(beta n) = (I + beta(n)A)(-1), for a maximal monotone operator A) in a real Hilbert space, under new sets of conditions on alpha(n) is an element of (0, 1) and beta(n) is an element of (0, infinity). 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. 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.
In this paper a proximal point algorithm (PPA) for maximal monotone operators with appropriate regularization parameters is considered. A strong convergence result for PPA is stated and proved under the general condit...
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In this paper a proximal point algorithm (PPA) for maximal monotone operators with appropriate regularization parameters is considered. A strong convergence result for PPA is stated and proved under the general condition that the error sequence tends to zero in norm. Note that Rockafellar (SIAM J Control Optim 14: 877-898, 1976) assumed summability for the error sequence to derive weak convergence of PPA in its initial form, and this restrictive condition on errors has been extensively used so far for different versions of PPA. Thus this Note provides a solution to a long standing open problem and in particular offers new possibilities towards the approximation of the minimum points of convex functionals.
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