Nonsingularity in matrix conic optimization induced by spectral norm via a smoothing metric projector

Matrix conic optimization induced by spectral norm (MOSN) has found important applications in many fields. This paper focus on the optimality conditions and perturbation analysis of the MOSN problem. The Karush–Kuhn–Tucker (KKT) conditions of the MOSN problem can be reformulated as a nonsmooth system via the metric projector over the cone. We show in this paper, the nonsingularity of the Clarke’s generalized Jacobian of the smoothing KKT system constructed by a smoothing metric projector, the strong regularity and the strong second‐order sufficient condition under constraint nondegeneracy are all equivalent. Moreover, this nonsingularity is used in several globally convergent smoothing Newton methods.