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

Xin-He Miao (), Yen-chi Roger Lin () and Jein-Shan Chen ()
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Xin-He Miao: Tianjin University
Yen-chi Roger Lin: National Taiwan Normal University
Jein-Shan Chen: National Taiwan Normal University

Journal of Optimization Theory and Applications, 2017, vol. 173, issue 3, No 18, 1066-1070

Abstract: 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.

Keywords: pth-order cone; Second-order cone; Inner product; Jordan algebras; 17C10; 52A07 (search for similar items in EconPapers)
Date: 2017
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DOI: 10.1007/s10957-017-1102-7

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