Preprint A754/2014
Combining stabilized SQP with the augmented Lagrangian algorithm
Alexey Izmailov | Mikhail Solodov | Evgeniy Uskov

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.