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Projections onto the Intersection of a One-Norm Ball or Sphere and a Two-Norm Ball or Sphere

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Abstract

This paper focuses on designing a unified approach for computing the projection onto the intersection of a one-norm ball or sphere and a two-norm ball or sphere. We show that the solutions of these problems can all be determined by the root of the same piecewise quadratic function. We make use of the special structure of the auxiliary function and propose a novel bisection algorithm with finite termination. We show that the proposed method possesses quadratic time worst-case complexity. The efficiency of the proposed algorithm is demonstrated in numerical experiments, which show the proposed method has linear time complexity in practice.

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Acknowledgements

Hongying Liu was supported by the National Natural Science Foundation of China under Grant 11822103 and Hao Wang by the Young Scientists Fund of the National Natural Science Foundation of China under Grant 12001367.

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Authors and Affiliations

  1. School of Mathematical Sciences, Beihang University, Beijing, 100083, China

    Hongying Liu & Mengmeng Song

  2. School of Information Science and Technology, ShanghaiTech University, Shanghai, 201210, China

    Hao Wang

Authors
  1. Hongying Liu
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  2. Hao Wang
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  3. Mengmeng Song
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Correspondence to Hao Wang.

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Communicated by Juan Parra.

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Liu, H., Wang, H. & Song, M. Projections onto the Intersection of a One-Norm Ball or Sphere and a Two-Norm Ball or Sphere. J Optim Theory Appl 187, 520–534 (2020). https://doi.org/10.1007/s10957-020-01766-y

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  • DOI: https://doi.org/10.1007/s10957-020-01766-y

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