  In this paper, we accomplish a unified convergence analysis of a second-order
method of multipliers (i.e., a second-order augmented Lagrangian method) for
solving the conventional nonlinear conic optimization problems.Specifically,
the algorithm that we investigated incorporates a specially designed nonsmooth
(generalized) Newton step to furnish a second-order update rule for the
multipliers.We first show in a unified fashion that under a few abstract
assumptions, the proposed method is locally convergent and possesses a
(nonasymptotic) superlinear convergence rate, even though the penalty parameter
is fixed and/or the strict complementarity fails.Subsequently, we demonstrate
that, for the three typical scenarios, i.e., the classic nonlinear programming,
the nonlinear second-order cone programming, and the nonlinear semidefinite
programming, these abstract assumptions are nothing but exactly the
implications of the iconic sufficient conditions that were assumed for
establishing the Q-linear convergence rates of the method of multipliers
without assuming the strict complementarity.
