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A Note on the Paper “The Algebraic Structure of the Arbitrary-Order Cone”

Author

Listed:
  • Xin-He Miao

    (Tianjin University)

  • Yen-chi Roger Lin

    (National Taiwan Normal University)

  • Jein-Shan Chen

    (National Taiwan Normal University)

Abstract

In this short paper, we look into a conclusion drawn by Alzalg (J Optim Theory Appl 169:32–49, 2016). We think the conclusion drawn in the paper is incorrect by pointing out three things. First, we provide a counterexample that the proposed inner product does not satisfy bilinearity. Secondly, we offer an argument why a pth-order cone cannot be self-dual under any reasonable inner product structure on $$\mathbb {R}^n$$ R n . Thirdly, even under the assumption that all elements operator commute, the inner product becomes an official inner product and the arbitrary-order cone can be shown as a symmetric cone, we think this condition is still unreasonable and very stringent so that the result can only be applied to very few cases.

Suggested Citation

  • Xin-He Miao & Yen-chi Roger Lin & Jein-Shan Chen, 2017. "A Note on the Paper “The Algebraic Structure of the Arbitrary-Order Cone”," Journal of Optimization Theory and Applications, Springer, vol. 173(3), pages 1066-1070, June.
    • Handle: RePEc:spr:joptap:v:173:y:2017:i:3:d:10.1007_s10957-017-1102-7
    DOI: 10.1007/s10957-017-1102-7
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    References listed on IDEAS

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    1. Xin-He Miao & Yu-Lin Chang & Jein-Shan Chen, 2017. "On merit functions for p-order cone complementarity problem," Computational Optimization and Applications, Springer, vol. 67(1), pages 155-173, May.
    2. Baha Alzalg, 2016. "The Algebraic Structure of the Arbitrary-Order Cone," Journal of Optimization Theory and Applications, Springer, vol. 169(1), pages 32-49, April.
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