A Relaxed Quasinormality Condition and the Boundedness of Dual Augmented Lagrangian Sequences

Abstract

Global convergence of augmented Lagrangian methods to a first-order stationary point is well known to hold under considerably weak constraint qualifications. In particular, several constant rank-type conditions have been introduced for this purpose which turned out to be relevant also beyond this scope. In this paper we show that in fact under these conditions subsequences of approximate Lagrange multipliers associated with accumulation points generated by the algorithm remains bounded. This important stability property is associated with both the practical effectiveness of the algorithm and its computational complexity. In order to obtain this result we introduce a relaxed version of the quasinormality constraint qualification which adequately treats equality constraints by means of informative Lagrange multipliers, a topic that has been extensively studied.