  This paper mainly concerns with the primal superlinear convergence of the
quasi-Newton sequential quadratic programming (SQP) method for piecewise
linear-quadratic composite optimization problems. We show that the latter
primal superlinear convergence can be justified under the noncriticality of
Lagrange multipliers and a version of the Dennis-More condition. Furthermore,
we show that if we replace the noncriticality condition with the second-order
sufficient condition, this primal superlinear convergence is equivalent with an
appropriate version of the Dennis-More condition. We also recover Bonnans'
result in [1] for the primal-dual superlinear of the basic SQP method for this
class of composite problems under the second-order sufficient condition and the
uniqueness of Lagrange multipliers. To achieve these goals, we first obtain an
extension of the reduction lemma for convex Piecewise linear-quadratic
functions and then provide a comprehensive analysis of the noncriticality of
Lagrange multipliers for composite problems. We also establish certain primal
estimates for KKT systems of composite problems, which play a significant role
in our local convergence analysis of the quasi-Newton SQP method.
