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The Algebraic Structure of the Arbitrary-Order Cone

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Abstract

We study and analyze the algebraic structure of the arbitrary-order cones. We show that, unlike popularly perceived, the arbitrary-order cone is self-dual for any order greater than or equal to 1. We establish a spectral decomposition, consider the Jordan algebra associated with this cone, and prove that this algebra forms a Euclidean Jordan algebra with a certain inner product. We generalize some important notions and properties in the Euclidean Jordan algebra of the second-order cone to the Euclidean Jordan algebra of the arbitrary-order cone.

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Acknowledgments

The author thanks the anonymous referees for their valuable suggestions to improve the paper.

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  1. Department of Mathematics, Faculty of Science, The University of Jordan, Amman, 11942, Jordan

    Baha Alzalg

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  1. Baha Alzalg
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Correspondence to Baha Alzalg.

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Alzalg, B. The Algebraic Structure of the Arbitrary-Order Cone. J Optim Theory Appl 169, 32–49 (2016). https://doi.org/10.1007/s10957-016-0878-1

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  • DOI: https://doi.org/10.1007/s10957-016-0878-1

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