Strong convergence of a proximal point algorithm with bounded error sequence |
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Authors: | Oganeditse A Boikanyo Gheorghe Moroşanu |
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Institution: | 1. Department of Mathematics, University of Botswana, Private Bag 00704, Gaborone, Botswana 2. Department of Mathematics and its Applications, Central European University, Nador u. 9, Budapest, 1051, Hungary
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Abstract: | Given any maximal monotone operator ${A: D(A)\subset H \rightarrow 2^H}$ in a real Hilbert space H with ${A^{-1}(0) \ne \emptyset}$ , it is shown that the sequence of proximal iterates ${x_{n+1}=(I+\gamma_n A)^{-1}(\lambda_n u+(1-\lambda_n)(x_n+e_n))}$ converges strongly to the metric projection of u on A ?1(0) for (e n ) bounded, ${\lambda_n \in (0,1)}$ with ${\lambda_n \to 1}$ and γ n > 0 with ${\gamma_n \to\infty}$ as ${n \to \infty}$ . In comparison with our previous paper (Boikanyo and Moro?anu in Optim Lett 4(4):635–641, 2010), where the error sequence was supposed to converge to zero, here we consider the classical condition that errors be bounded. In the case when A is the subdifferential of a proper convex lower semicontinuous function ${\varphi :H \to (-\infty,+ \infty]}$ , the algorithm can be used to approximate the minimizer of φ which is nearest to u. |
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