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

作者：Miao, Xin-He; Lin, Yen-chi Roger^*; Chen, Jein-Shan

来源：Journal of Optimization Theory and Applications, 2017, 173(3): 1066-1070.

DOI：10.1007/s10957-017-1102-7

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

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 . 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.

• 出版日期2017-6
• 单位天津大学