On constraint qualifications for non-relaxable sets and an augmented Lagrangian method

Abstract In this paper we consider an augmented Lagrangian method with general lower-level constraints, that is, where some of the constraints are penalized while others are kept as subproblem constraints. Motivated by some recent results on optimization problems on manifolds, we present a general theory of global convergence when a feasible approximate KKT point is found for the subproblems at each iteration. In particular, we formulate new constant rank constraint qualifications that do not require a constant rank assumption in a full dimensional neighborhood of the point of interest. We also formulate an appropriate quasinormality and relaxed-quasinormality conditions which guarantee boundedness of the dual sequences generated by the algorithm. These assumptions apply, in particular, to the current ALGENCAN implementation that keeps box constraints within the subproblems.