  We introduce a quadratically convergent semismooth Newton method for
nonlinear semidefinite programming that eliminates the need for the
subdifferential regularity, a common yet stringent requirement in existing
approaches. Our strategy revolves around identifying a single nonsingular
element within the B(ouligand)-subdifferential, thus avoiding the standard
requirement for nonsingularity across the entire subdifferential set, which is
often too restrictive for practical applications. The theoretical framework is
supported by the introduction of the weak second order condition (W-SOC) and
the weak strict Robinson constraint qualification (W-SRCQ). These conditions
not only guarantee the existence of a nonsingular element in the
subdifferential but also forge a primal-dual connection in linearly constrained
convex quadratic programming. The theoretical advancements further lay the
foundation for the algorithmic designing of a novel semismooth Newton method,
which integrates a correction step to address degenerate issues. Particularly,
this correction step ensures the local convergence as well as a
superlinear/quadratic convergence rate of the proposed method. Preliminary
numerical experiments corroborate our theoretical findings and underscore the
practical effectiveness of our method.
