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 \(\mathbb {R}^n\). 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.
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References
Alzalg, B.: The algebraic structure of the arbitrary-order cone. J. Optim. Theory Appl. 169, 32–49 (2016)
Ito, M., Lourenço, B.F.: The \(p\)-cone in dimension \(n \ge 3\) are not homogeneous when \(p \ne 2\). Optim. Online 1–11 (2016)
Rockafellar, R.: Convex Analysis. Princeton University Press, Princeton (1997)
Miao, X.-H., Chang, Y.-L., Chen, J.-S.: On merit functions for \(p\)-order cone complementarity problem. Comput. Optim. Appl. doi:10.1007/s10589-016-9889-y (2017)
Schmieta, S.H., Alizadeh, F.: Extension of primal-dual interior point methods to symmetric cones. Math. Program. Ser. A 96, 409–438 (2003)
Faraut, U., Korányi, A.: Analysis on Symmetric Cones. Oxford Mathematical Monographs, Oxford University Press, New York (1994)
Acknowledgements
The first author’s work is supported by National Natural Science Foundation of China (No. 11471241). The third author’s work is supported by Ministry of Science and Technology, Taiwan. The paragraph about that the only Euclidean Jordan algebra where all elements operator commute is \(\mathbb {R}^n\), suggested by one of the reviewers, is so valuable that the authors decide to include it. The authors would like to thank anonymous reviewers for suggestions, which improved the paper.
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Communicated by Sándor Zoltán Németh.
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Miao, XH., Lin, Yc.R. & Chen, JS. A Note on the Paper “The Algebraic Structure of the Arbitrary-Order Cone”. J Optim Theory Appl 173, 1066–1070 (2017). https://doi.org/10.1007/s10957-017-1102-7
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DOI: https://doi.org/10.1007/s10957-017-1102-7