Augmented Lagrangian methods applied to optimization problems with degenerate constraints, including problems with complementarity constraints
E.I. Uskov | Izmailov, A.F. | Solodov, M.V.
We consider global convergence properties of the augmented Lagrangian methods on problems with degenerate constraints, with a special emphasis on mathematical programs with complementarity constraints (MPCC). In the general case, we show convergence to stationary points of the problem under an error bound condition for the feasible set (which is weaker than constraint qualifications), assuming that the iterates have some modest features of approximate local minimizers of the augmented Lagrangian. For MPCC, we first argue that even weak forms of general constraint qualifications that are suitable for convergence of the augmented Lagrangian methods, such as the recently proposed relaxed positive linear dependence condition, should not be expected to hold and thus special analysis is needed. We next obtain a rather complete picture, showing that under the usual in this context MPCC-linear independence constraint qualification feasible accumulation points of the iterates are guaranteed to be C-stationary for MPCC (better than weakly stationary), but in general need not be M-stationary (hence, neither strongly stationary). However, strong stationarity is guaranteed if the generated dual sequence is bounded, which we show to be the typical numerical behaviour even though the multiplier set itself is unbounded. Experiments with the ALGENCAN augmented Lagrangian solver on the MacMPEC and DEGEN collections are reported, with comparisons to the SNOPT and filterSQP implementations of the SQP method, to the MINOS implementation of the linearly constrained Lagrangian method, and to the interior-point solvers IPOPT and KNITRO. We show that ALGENCAN is a very good option if one is primarily interested in robustness and quality of computed solutions.