Global convergence of augmented Lagrangian methods applied to optimization problems with degenerate constraints, including problems with complementarity constraints.

*(English)*Zbl 1274.90385The authors consider the mathematical programming problem where the objective function and the constraint mappings are smooth. The goal of this paper is to clarify the behavior of the augmented Lagrangian methods in the cases of degenerate constraints. The degeneracy means violation of (more-or-less) standard constraint qualifications at some (or all) feasible points. An important case of degenerate problems is the class of the mathematical programs with complementarity constraints (MPCC). In the general case, the authors proved the convergence of the augmented Lagrangian method to stationary points of the problem, assuming that an error bound holds for the fesible set (which is weaker than constraint qualifications, including the relaxed positive linear dependance condition) and assuming that the iterates have some modest features of approximate local minimizers of the augmented Lagrangian. For MPCC, under the MPCC-linear independence constraint qualification, it is shown that accumulation points of the augmented Lagrangian iterates are guaranteed to be C-stationary, and they are strongly stationary if the generated dual sequence is bounded. Numerical results with the ALGENCAN augmented Lagrangian solver on the MacMPEC and DEGEN collections are presented.

Reviewer: Nada Djuranović-Miličić (Belgrade)

##### MSC:

90C30 | Nonlinear programming |

90C33 | Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming) |

90C55 | Methods of successive quadratic programming type |

65K05 | Numerical mathematical programming methods |