Here, we present two variants of the augmented Lagrangian method that use the conjugate gradient method in the inner loop. We start with the asymptotically exact augmented Lagrangian method, controlling the precision of the auxiliary problem’s solutions by a user-defined forcing sequence. The regularization parameter controls the convergence rate of the outer loop. The second method is SMALE (semi-monotonic augmented Lagrangian for equality constraints). The inner loop precision is regulated by the increase of the Lagrangian. The unique theoretical results concerning SMALE include a bound on the number of iterations necessary to find an approximate solution with prescribed error and independent of the conditioning of constraints. The performance of SMALE can be improved by preconditioning that preserves the gap in the spectrum. The algorithms accept dependent equality constraints.