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Exact convergence rates of alternating projections for nontransversal intersections

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Abstract

We consider the convergence rate of the alternating projection method for the nontransversal intersection of a semialgebraic set and a linear subspace. For such an intersection, the convergence rate is known as sublinear in the worst case. We study the exact convergence rate for a given semialgebraic set and an initial point, and investigate when the convergence rate is linear or sublinear. As a consequence, we show that the exact rates are expressed by multiplicities of the defining polynomials of the semialgebraic set, or related power series in the case that the linear subspace is a line, and we also decide the convergence rate for given data by using elimination theory. Our methods are also applied to give upper bounds for the case that the linear subspace has the dimension more than one. The upper bounds are shown to be tight by obtaining exact convergence rates for a specific semialgebraic set, which depend on the initial points.

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Acknowledgements

The first author was supported by JSPS KAKENHI Grant Number JP17K18726. The second author was supported by JSPS KAKENHI Grant Number JP19K03631. The third author was supported by JSPS KAKENHI Grant Number JP20K11696 and ERATO HASUO Metamathematics for Systems Design Project (No. JPMJER1603), JST.

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Authors and Affiliations

  1. Institute of Mathematics for Industry, Kyushu University, 744 Motooka, Nishi-ku, Fukuoka, 819-0395, Japan

    Hiroyuki Ochiai & Hayato Waki

  2. Graduate School of Marine Science and Technology, Tokyo University of Marine Science and Technology, Etchujima 2-1-8, Koto-ku, Tokyo, 135-8533, Japan

    Yoshiyuki Sekiguchi

Authors
  1. Hiroyuki Ochiai
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  2. Yoshiyuki Sekiguchi
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  3. Hayato Waki
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Correspondence to Yoshiyuki Sekiguchi.

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Ochiai, H., Sekiguchi, Y. & Waki, H. Exact convergence rates of alternating projections for nontransversal intersections. Japan J. Indust. Appl. Math. 41, 57–83 (2024). https://doi.org/10.1007/s13160-023-00584-9

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  • DOI: https://doi.org/10.1007/s13160-023-00584-9

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