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. We define notions of the algebraic structure of the nonconvex second-order cone, and show that its ambient space is a commutative power-associative magma whose elements always have real eigenvalues; this is remarkable because it is not the case for arbitrary Jordan algebras. We will also find that the magma of this nonconvex cone is rankly independent of its dimension; this is also remarkable because it is not the case for algebras of arbitrary convex cones. Even more remarkably, we prove that the nonconvex second-order cone equals the cone of squares of its magma; 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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View The Algebraic Structure of the Nonconvex Second-Order Cone