  Strong variational sufficiency is a newly proposed property, which turns out
to be of great use in the convergence analysis of multiplier methods. However,
what this property implies for non-polyhedral problems remains a puzzle. In
this paper, we prove the equivalence between the strong variational sufficiency
and the strong second order sufficient condition (SOSC) for nonlinear
semidefinite programming (NLSDP), without requiring the uniqueness of
multiplier or any other constraint qualifications. Based on this
characterization, the local convergence property of the augmented Lagrangian
method (ALM) for NLSDP can be established under strong SOSC in the absence of
constraint qualifications. Moreover, under the strong SOSC, we can apply the
semi-smooth Newton method to solve the ALM subproblems of NLSDP as the positive
definiteness of the generalized Hessian of augmented Lagrangian function is
satisfied.
