  Second-order sufficient conditions for local optimality have been playing an
important role in local convergence analysis of optimization algorithms. In
this paper, we demonstrate that this condition alone suffices to justify the
linear convergence of the primal-dual sequence, generated by the augmented
Lagrangian method for piecewise linear-quadratic composite optimization
problems, even when the Lagrange multiplier in this class of problems is not
unique. Furthermore, we establish the equivalence between the second-order
sufficient condition and the quadratic growth condition of the augmented
Lagrangian problem for this class of composite optimization problems.

Optimization and Control