**Keywords:**

For an optimization problem with general equality and inequality constraints,

we propose an algorithm which uses subproblems of the stabilized SQP (sSQP) type

for approximately solving subproblems of the augmented Lagrangian method. The

motivation is to take advantage of the well-known robust behavior of the augmented

Lagrangian algorithm, including on problems with degenerate constraints, and at the

same time try to reduce the overall algorithm locally to sSQP (which gives fast local

convergence rate under weak assumptions). Specifically, the algorithm first verifies

whether the primal-dual sSQP step (with unit stepsize) makes good progress towards

decreasing the violation of optimality conditions for the original problem, and if

so, makes this step. Otherwise, the primal part of the sSQP direction is used for

linesearch that decreases the augmented Lagrangian, keeping the multiplier estimate

fixed for the time being. The overall algorithm has reasonable global convergence

guarantees, and inherits strong local convergence rate properties of sSQP under the same

weak assumptions. Numerical results on degenerate problems and comparisons with some

alternatives are reported.