Volume: 387

Subspace properties are presented for the trust-region subproblem that appears in the Augmented Lagrangian-Trust-Region method recently proposed by Wang and Yuan (2015). Specifically, when the approximate Lagrangian Hessians are updated by suitable quasi-Newton formulas, we show that any solution of the corresponding kth subproblem belongs to the subspace spanned by all gradient vectors of the objective and of the constraints computed up to iteration k. From this result, a subspace version of the referred method is proposed for large-scale equality constrained optimization problems. The subspace method is suitable to problems in which the number of constraints is much lower than the number of variables.