The Algebraic Structure of the Arbitrary-Order Cone

Alzalg, Baha

doi:10.1007/s10957-016-0878-1

Cited by 1 publication

(7 citation statements)

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“…, n. This is a reasonable assumption on the inner products to make, since we will almost never take arbitrary frames in R n to define a pth-order cone. Also we note that the "inner product" •, • p appeared in [1] satisfies the property (2).…”

Section: Example 21 For Anymentioning

confidence: 86%

“…We think this condition is unreasonable and very stringent by elaborating the reason and counterexample. To sum up, we think the conclusion drawn in the paper [1] is very limited.…”

Section: Introductionmentioning

confidence: 85%

“…Alzalg argues that x, y p defined as in (1) is a new inner product on R n , and the pth-order cone becomes symmetric under this new product. The pth-order cone in R n is defined as…”

Section: A Type Of Inner Productmentioning

confidence: 99%

“…In the recent paper [1], Alzalg claims that the arbitrary-order cone is a symmetric cone for any order greater than or equal to 1, that is, the cone is homogeneous and self-dual. Ito and Lourenço [2] showed that the pth-order cone in dimension n ≥ 3 are not homogeneous unless p = 2.…”

Section: Introductionmentioning

confidence: 99%

“…In this short note, we show that the arbitrary-order cone is not self-dual for any order other than 2. In particular, we provide a counterexample indicating that the inner product defined in [1] is indeed not an inner product because it does not satisfy the bilinearity. We offer an argument why a pth-order cone cannot be self-dual under any reasonable inner product structure on R n .…”

Section: Introductionmentioning

confidence: 99%

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

Miao

Lin

Chen

2017

J Optim Theory Appl

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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

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Section: Example 21 For Anymentioning

confidence: 86%

“…We think this condition is unreasonable and very stringent by elaborating the reason and counterexample. To sum up, we think the conclusion drawn in the paper [1] is very limited.…”

Section: Introductionmentioning

confidence: 85%

“…Alzalg argues that x, y p defined as in (1) is a new inner product on R n , and the pth-order cone becomes symmetric under this new product. The pth-order cone in R n is defined as…”

Section: A Type Of Inner Productmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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