Tight Coefficients of Averaged Operators via Scaled Relative Graph

Xinmeng Huang University of Science and Technology of China Ernest K. Ryu University of California, Los Angeles Wotao Yin University of California, Los Angeles

Optimization and Control mathscidoc:2004.27016

Journal of Mathematical Analysis and Applications, 2020.5
Many iterative methods in optimization are fixed-point iterations with averaged operators. As such methods converge at an O(1/k) rate with the constant determined by the averagedness coefficient, establishing small averagedness coefficients for operators is of broad interest. In this paper, we show that the averagedness coefficients of the composition of averaged operators by Ogura and Yamada (Numer Func Anal Opt 32(1–2):113–137, 2002) and the threeoperator splitting by Davis and Yin (Set Valued Var Anal 25(4):829–858, 2017) are tight. The analysis relies on the scaled relative graph, a geometric tool recently proposed by Ryu, Hannah, and Yin (arXiv:1902.09788, 2019).
Averaged operator, Composition of operators, Nonexpansive operator, Euclidean geometry, Three operators
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@inproceedings{xinmeng2020tight,
  title={Tight Coefficients of Averaged Operators via Scaled Relative Graph},
  author={Xinmeng Huang, Ernest K. Ryu, and Wotao Yin},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20200427085035212943656},
  booktitle={Journal of Mathematical Analysis and Applications},
  year={2020},
}
Xinmeng Huang, Ernest K. Ryu, and Wotao Yin. Tight Coefficients of Averaged Operators via Scaled Relative Graph. 2020. In Journal of Mathematical Analysis and Applications. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20200427085035212943656.
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