Local convergence of augmented Lagrangian methods under the second-order sufficient optimality condition

DAMIÁN FERNÁNDEZ; MIKHAIL SOLODOV

Córdoba

Congreso; IV Congress of Latin American Mathematicians; 2012

Universidad Nacional de Córdoba

We establish local convergence and rate of convergenceof the classical augmented Lagrangian algorithm under thesole assumption that the dual starting point is close to amultiplier satisfying the second-order sufficient optimality condition. In particular, no constraint qualificationsof any kind are needed. We prove primal-dual $Q$-linearconvergence rate, which becomes superlinear if the parameters are allowedto go to infinity. Both exact and inexact solutions of subproblemsare considered. In the exact case, we further show that theprimal convergence rate is of the same $Q$-order as the primal-dual rate.Previous literature on the subject required the second-order sufficient optimality condition, the linear independence constraint qualification andeither the strict complementarity assumption or a strongerversion of the second-order sufficient condition. If the stronger assumptions are introduced, our analysis recovers the classical results thatallow the initial multiplier estimate to be far from the optimal oneat the expense of proportionally increasing the penalty parameters.Finally, we show that under our assumptions one of the popular rulesof controlling the penalty parameters ensures their boundedness.