Strong variational sufficiency for nonlinear semidefinite programming and its implications

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 nonpolyhedral 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 the 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 the strong SOSC in the absence of constraint qualifications. Moreover, under the strong SOSC, we can apply the semismooth Newton method to solve the ALM subproblems of NLSDP because the positive definiteness of the generalized Hessian of augmented Lagrangian function is satisfied.