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On the Finite Convergence of a Projected Cutter Method

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Abstract

The subgradient projection iteration is a classical method for solving a convex inequality. Motivated by works of Polyak and of Crombez, we present and analyze a more general method for finding a fixed point of a cutter, provided that the fixed point set has nonempty interior. Our assumptions on the parameters are more general than existing ones. Various limiting examples and comparisons are provided.

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Notes

  1. In fact, \(U_r\) is not even a relaxed cutter in the sense of [2, Definition 2.1.30].

  2. \(U_{r,\eta }\) can also be called a generalized relaxation of \(T\) with relaxation parameter \(\eta \); see [2, Definition 2.4.1].

  3. We note that item (iv) can also be deduced from [2, (2.27)] with \(\lambda = (r+\Vert x-Tx\Vert )/\Vert x-Tx\Vert \), \(z=y\), and \(\delta =r\) in [2, Proposition 2.1.41]. This observation, as well as a similar one for (v), is due to a referee.

  4. This observation is a due to a referee.

  5. If we replace Fréchet differentiability by mere continuity, then we may consider a selection of the subdifferential operator \(\partial f\) instead.

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Acknowledgments

The authors thank two anonymous referees for careful reading, constructive comments, and for bringing additional references to our attention. The authors also thank Jeffrey Pang for helpful discussions and for pointing out additional references. HHB was partially supported by a Discovery Grant and an Accelerator Supplement of the Natural Sciences and Engineering Research Council of Canada (NSERC) and by the Canada Research Chair Program. CW is partially supported by a Grant from Shanghai Municipal Commission for Science and Technology (13ZR1455500). XW is partially supported by a Discovery Grant of NSERC. JX is partially supported by NSERC Grants of HHB and XW.

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Authors and Affiliations

  1. Department of Mathematics, University of British Columbia, Kelowna, BC, V1V 1V7, Canada

    Heinz H. Bauschke, Xianfu Wang & Jia Xu

  2. Department of Mathematics, Shanghai Maritime University, Shanghai, China

    Caifang Wang

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  1. Heinz H. Bauschke
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  2. Caifang Wang
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  3. Xianfu Wang
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Correspondence to Heinz H. Bauschke.

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Bauschke, H.H., Wang, C., Wang, X. et al. On the Finite Convergence of a Projected Cutter Method. J Optim Theory Appl 165, 901–916 (2015). https://doi.org/10.1007/s10957-014-0659-7

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