The nonconvex second-order cone (nonconvex SOC for short) is a nonconvex extension to the convex second-order cone, in the sense that it consists of any vector divided into two sub-vectors for which the Euclidean norm of the first sub-vector is at least as large as the Euclidean norm of the second sub-vector. This cone can be used to reformulate nonconvex quadratic programs in conic format and can arise in real-world applications. In this paper, spectral scalar and vector-valued functions associated with the nonconvex SOC are defined analogously to the corresponding functions associated with the convex second-order cone. We present several properties and key characteristics of the nonconvex SOC-related functions. The results in this paper are useful for developing and analyzing solution methods for solving optimization problems over the nonconvex SOC and their complementarity problems.
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