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

Authors:

Xin-He Miao,

Yen-Chi Roger Lin,

Jein-Shan ChenAuthors Info & Claims

Journal of Optimization Theory and Applications, Volume 173, Issue 3

Pages 1066 - 1070

https://doi.org/10.1007/s10957-017-1102-7

Published: 01 June 2017 Publication History

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 $$\mathbb {R}^n$$Rn. 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.

References

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Alzalg, B.: The algebraic structure of the arbitrary-order cone. J. Optim. Theory Appl. 169, 32---49 (2016)

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Ito, M., Lourenço, B.F.: The $$p$$p-cone in dimension $$n \ge 3$$n¿3 are not homogeneous when $$p \ne 2$$p¿2. Optim. Online 1---11 (2016)

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Rockafellar, R.: Convex Analysis. Princeton University Press, Princeton (1997)

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Miao, X.-H., Chang, Y.-L., Chen, J.-S.: On merit functions for $$p$$p-order cone complementarity problem. Comput. Optim. Appl. (2017)

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cover image Journal of Optimization Theory and Applications

Journal of Optimization Theory and Applications Volume 173, Issue 3

June 2017

344 pages

ISSN:0022-3239

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Plenum Press

United States

Publication History

Published: 01 June 2017

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Abstract

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The Algebraic Structure of the Arbitrary-Order Cone

An Interior Point Cutting Plane Method for the Convex Feasibility Problem with Second-Order Cone Inequalities

Integer Points in Arbitrary Convex Cones: The Case of the PSD and SOC Cones

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