Nonsingularity of Karush-Kuhn-Tucker system in a class of linear matrix conic optimization problems

It is known that the Karush-Kuhn-Tucker (KKT) conditions of matrix conic optimization problem can be reformulated as a system via the metric projector over the cone of matrices. We consider a class of linear matrix conic optimization problems, where the cone is the epigraph of an orthogonally invariant norm induced by a symmetric function, in the space of symmetric matrices. We show that the nonsigularity of Jacobian of the KKT system is equivalent to the primal constraint nondegeneracy and the second-order sufficient optimality conditions. Moreover, we prove that, for the second-order cone defined by the epigraph of Frobenius norm, the primal second-order sufficient optimality conditions are equivalent to the dual constraint nondegeneracy condition.