Local Convergence of Exact and Inexact Augmented Lagrangian Methods under the Second-Order Sufficient Optimality Condition

Abstract

Abstract. We establish local convergence and rate of convergence of the classical augmented Lagrangian algorithm under the sole assumption that the dual starting point is close to a multiplier satisfying the second-order sufficient optimality condition. In particular, no constraint qualifications of any kind are needed. Previous literature on the subject required, in addition, the linear independence constraint qualification and either the strict complementarity assumption or a stronger version of the second-order sufficient condition. That said, the classical results allow the initial multiplier estimate to be far from the optimal one, at the expense of proportionally increasing the threshold value for the penalty parameters. Although our primary goal is to avoid constraint qualifications, if the stronger assumptions are introduced, then starting points far from the optimal multiplier are allowed within our analysis as well. Using only the second-order sufficient optimality condition, for penalty parameters large enough we prove primal-dual Q-linear convergence rate, which becomes superlinear if the parameters are allowed to go to infinity. Both exact and inexact solutions of subproblems are considered. In the exact case, we further show that the primal convergence rate is of the same Q-order as the primal-dual rate. Previous assertions for the primal sequence all had to do with the weaker R-rate of convergence and required the stronger assumptions cited above. Finally, we show that under our assumptions one of the popular rules of controlling the penalty parameters ensures their boundedness.

Global Convergence of Augmented Lagrangian Methods Applied to Optimization Problems with Degenerate Constraints, Including Problems with Complementarity Constraints

“…It should also be mentioned that IPOPT-C [49] apparently is not supported by any global convergence theory. AL-GENCAN definitely outperforms its competitors in terms of major iterations count, which is the sign of a higher convergence rate, in agreement with the local rate of convergence results in [20] that allow any kind of degeneracy. On the other hand, the cost of (especially late) iterations of ALGENCAN is rather high, and so its convergence rate does not translate into saved CPU time.…”

A Globally Convergent Stabilized SQP Method

“…In particular, Assumptions 3.1-3.3 do not preclude the possibility that problem (NP) is infeasible. Recent work indicates that the iterates of the stabilized SQP subproblem exhibit superlinear convergence under mild conditions (see, e.g., [13,14,35,36]). In particular, neither strict complementarity nor a constraint qualification are required.…”

A note on upper Lipschitz stability, error bounds, and critical multipliers for Lipschitz-continuous KKT systems

“…For example, SOSC (12) was the only assumption needed to prove local convergence of the stabilized sequential quadratic programming method in [5] and of the augmented Lagrangian algorithm in [6], with the error bound (8) playing a key role. When there are equality constraints only, the error bound itself (equivalently, noncriticality of the multiplier) is enough in the case of stabilized sequential quadratic programming [17].…”