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The Nonconvex Second-Order Cone: Algebraic Structure Toward Optimization

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Abstract

This paper explores the nonconvex second-order cone as a nonconvex conic extension of the known convex second-order cone in optimization, as well as a higher-dimensional conic extension of the known causality cone in relativity. The nonconvex second-order cone can be used to reformulate nonconvex quadratic programming and nonconvex quadratically constrained quadratic program in conic format. The cone can also arise in real-world applications, such as facility location problems in optimization when some existing facilities are more likely to be closer to new facilities than other existing facilities. We define notions of the algebraic structure of the nonconvex second-order cone and show that its ambient space is commutative and power-associative, wherein elements always have real eigenvalues; this is remarkable because it is not the case for arbitrary Jordan algebras. We will also find that the ambient space of this nonconvex cone is rank-independent of its dimension; this is also notable because it is not the case for algebras of arbitrary convex cones. What is more noteworthy is that we prove that the nonconvex second-order cone equals the cone of squares of its ambient space; this is not the case for all non-Euclidean Jordan algebras. Finally, numerous algebraic properties that already exist in the framework of the convex second-order cone are generalized to the framework of the nonconvex second-order cone.

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Acknowledgements

We thank the two anonymous referees for their careful reading of this paper and their suggestions. These comments helped to significantly improve the presentation of this paper. In addition, we express our appreciation to one of the referees for pointing out a small gap in one of the proofs in an earlier draft that resulted in an incomplete proof of Lemma 4.2. This has now been addressed and completed in an updated version of the paper.

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  1. The University of Jordan, Amman, 11942, Jordan

    Baha Alzalg & Lilia Benakkouche

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  1. Baha Alzalg
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  2. Lilia Benakkouche
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Correspondence to Baha Alzalg.

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Communicated by Alper Yildirim.

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Alzalg, B., Benakkouche, L. The Nonconvex Second-Order Cone: Algebraic Structure Toward Optimization. J Optim Theory Appl (2024). https://doi.org/10.1007/s10957-024-02406-5

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