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

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 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 Mathematics Subject Classification 17C10 • 52A07 Communicated by Sándor Zoltán Németh.B Yen-chi Roger Lin

The $p$-cones in dimension $n \geq 3$ are not homogeneous when $p\neq 2$

“…In the same article, it was claimed that L n p is homogeneous, thus showing that it is a symmetric cone under an appropriate inner product. For a discussion of its flaws, we refer to the paper by Miao, Lin, Chen [8]. The result we prove here implies, in particular, that for n ≥ 3 and p = 2, L n p is never a symmetric cone.…”

“…The authors would like to thank Prof. Chen for kindly agreeing to send us a preliminary version of [8]. The authors would also like to thank an anonymous referee for many valuable suggestions that improved the presentation of this paper.…”

“…This, however, does not rule out the possibility of L n p being self-dual under an appropriate inner product, which also seems to be an unsettled problem. We remark that it can be shown that L n p is not self-dual under "reasonable" inner products, see Section 3 in [8].…”

The automorphism group and the non-self-duality of p-cones

“…Now, although we know that L n+1 p is not homogeneous for p = 2 and n ≥ 2, it still remains to investigate whether L n+1 p could become self-dual under an appropriate inner product. This question was partly discussed by Miao, Lin and Chen in [9], where they showed that a p-cone (again, p = 2, n ≥ 2) is not self-dual under an inner product induced by a diagonal matrix. The results described here show, in particular, that no inner product can make L n+1 p self-dual, for p = 2, n ≥ 2.…”