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

Xin He Miao, Yen chi Roger Lin*, Jein Shan Chen

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

3 Citations (Scopus)


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 Rn. 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.

Original languageEnglish
Pages (from-to)1066-1070
Number of pages5
JournalJournal of Optimization Theory and Applications
Issue number3
Publication statusPublished - 2017 Jun 1


  • Inner product
  • Jordan algebras
  • Second-order cone
  • pth-order cone

ASJC Scopus subject areas

  • Management Science and Operations Research
  • Control and Optimization
  • Applied Mathematics


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