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A note on alternating projections for ill-posed semidefinite feasibility problems

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Abstract

We observe that Sturm’s error bounds readily imply that for semidefinite feasibility problems, the method of alternating projections converges at a rate of \(\mathcal {O}\Big (k^{-\frac{1}{2^{d+1}-2}}\Big )\), where d is the singularity degree of the problem—the minimal number of facial reduction iterations needed to induce Slater’s condition. Consequently, for almost all such problems (in the sense of Lebesgue measure), alternating projections converge at a worst-case rate of \(\mathcal {O}\Big (\frac{1}{\sqrt{k}}\Big )\).

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Acknowledgments

We thank Levent Tunçel for insightful discussions, and the two anonymous referees for their comments and suggestions.

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Authors and Affiliations

  1. Department of Mathematics, University of Washington, Seattle, WA, 98195, USA

    Dmitriy Drusvyatskiy

  2. Department of Applied Mathematics, University of New South Wales, Sydney, 2052, Australia

    Guoyin Li

  3. Department of Combinatorics and Optimization, University of Waterloo, Waterloo, ON, N2L 3G1, Canada

    Henry Wolkowicz

Authors
  1. Dmitriy Drusvyatskiy
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  2. Guoyin Li
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  3. Henry Wolkowicz
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Correspondence to Dmitriy Drusvyatskiy.

Additional information

Research of G. Li was supported by an ARC Future Fellowship. Research of H. Wolkowicz was supported by The Natural Sciences and Engineering Research Council of Canada and AFOSR.

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Drusvyatskiy, D., Li, G. & Wolkowicz, H. A note on alternating projections for ill-posed semidefinite feasibility problems. Math. Program. 162, 537–548 (2017). https://doi.org/10.1007/s10107-016-1048-9

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