Constraint Nondegeneracy, Strong Regularity, and Nonsingularity in Semidefinite Programming

It is known that the Karush-Kuhn-Tucker (KKT) conditions of semidefinite programming can be reformulated as a nonsmooth system via the metric projector over the cone of symmetric and positive semidefinite matrices. We show in this paper that the primal and dual constraint nondegeneracies, the strong regularity, the nonsingularity of the B-subdifferential of this nonsmooth system, and the nonsingularity of the corresponding Clarke’s generalized Jacobian, which is the convex hull of the B-subdifferential, at a KKT point are all equivalent. Moreover, we prove the equivalence between each of these conditions and the nonsingularity of the B-subdifferential (or Clarke’s generalized Jacobian) of the smoothed counterpart of this nonsmooth system used in several smoothing Newton methods. In particular, we establish the quadratic convergence of these methods under the primal and dual constraint nondegeneracies, but without the strict complementarity.