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A supplement to a regularization method for the proximal point algorithm

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The purpose of this paper is to show that the iterative scheme recently studied by Xu (J Glob Optim 36(1):115–125, 2006) is the same as the one studied by Kamimura and Takahashi (J Approx Theory 106(2):226–240, 2000) and to give a supplement to these results. With the new technique proposed by Maingé (Comput Math Appl 59(1):74–79, 2010), we show that the convergence of the iterative scheme is established under another assumption. It is noted that if the computation error is zero or the approximate computation is exact, our new result is a genuine generalization of Xu’s result and Kamimura–Takahashi’s result.

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  1. Goebel K., Kirk W.A.: Topics in Metric Fixed Point Theory. Cambridge Studies in Advanced Mathematics, vol. 28. Cambridge University Press, Cambridge (1990)

    Book  Google Scholar 

  2. Kamimura S., Takahashi W.: Approximating solutions of maximal monotone operators in Hilbert spaces. J. Approx. Theory 106(2), 226–240 (2000)

    Article  Google Scholar 

  3. Maingé P.-E.: The viscosity approximation process for quasi-nonexpansive mappings in Hilbert spaces. Comput. Math. Appl. 59(1), 74–79 (2010)

    Article  Google Scholar 

  4. Minty G.J.: Monotone (nonlinear) operators in Hilbert space. Duke Math. J. 29, 341–346 (1962)

    Article  Google Scholar 

  5. Song Y., Changsen Y.: A note on a paper “A regularization method for the proximal point algorithm”. J. Glob. Optim. 43(1), 171–174 (2009)

    Article  Google Scholar 

  6. Takahashi W.: Introduction to Nonlinear and Convex Analysis. Yokohama Publishers, Yokohama (2009)

    Google Scholar 

  7. Xu H.-K.: Iterative algorithms for nonlinear operators. J. Lond. Math. Soc. 66(2), 240–256 (2002)

    Article  Google Scholar 

  8. Xu H.-K.: A regularization method for the proximal point algorithm. J. Glob. Optim. 36(1), 115–125 (2006)

    Article  Google Scholar 

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Correspondence to Satit Saejung.

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Saejung, S. A supplement to a regularization method for the proximal point algorithm. J Glob Optim 56, 121–129 (2013).

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