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On Convergence of the Arrow–Hurwicz Method for Saddle Point Problems

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Abstract

The Arrow–Hurwicz method is an inexact version of the Uzawa method; it has been widely applied to solve various saddle point problems in different areas including many fundamental image processing problems. It is also the basis of a number of important algorithms such as the extragradient method and the primal–dual hybrid gradient method. Convergence of the classic Arrow–Hurwicz method, however, is known only when some more restrictive conditions are additionally assumed, such as strong convexity of the functions or some demanding requirements on the step sizes. In this short note, we show by very simple counterexamples that the classic Arrow–Hurwicz method with any constant step size is not necessarily convergent for solving generic convex saddle point problems, including some fundamental cases such as the canonical linear programming model and the bilinear saddle point problem. This result plainly fathoms the convergence understanding of the Arrow–Hurwicz method and retrospectively validates the rationale of studying its convergence under various additional conditions in image processing literature.

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Funding

This study was supported by National Natural Science Foundation of China (Grant No. 11871029) and Research Grants Council, University Grants Committee (Grant No. 12302318).

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Author notes

  1. Shengjie Xu and Xiaoming Yuan have contributed equally to this work.

Authors and Affiliations

  1. Department of Mathematics, Nanjing University, Nanjing, China

    Bingsheng He

  2. Department of Mathematics, Harbin Institute of Technology, Harbin, China

    Shengjie Xu

  3. Department of Mathematics, Southern University of Science and Technology, Shenzhen, China

    Shengjie Xu

  4. Department of Mathematics, The University of Hong Kong, Pok Fu Lam, Hong Kong

    Xiaoming Yuan

Authors
  1. Bingsheng He
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  2. Shengjie Xu
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  3. Xiaoming Yuan
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Correspondence to Xiaoming Yuan.

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He, B., Xu, S. & Yuan, X. On Convergence of the Arrow–Hurwicz Method for Saddle Point Problems. J Math Imaging Vis 64, 662–671 (2022). https://doi.org/10.1007/s10851-022-01089-9

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  • DOI: https://doi.org/10.1007/s10851-022-01089-9

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