The Rate of Convergence of Augmented Lagrangian Method for Minimax Optimization Problems with Equality Constraints

The augmented Lagrangian function and the corresponding augmented Lagrangian method are constructed for solving a class of minimax optimization problems with equality constraints. We prove that, under the linear independence constraint qualification and the second-order sufficiency optimality condition for the lower level problem and the second-order sufficiency optimality condition for the minimax problem, for a given multiplier vector mu, the rate of convergence of the augmented Lagrangian method is linear with respect to parallel to mu - mu*parallel to and the ratio constant is proportional to 1/c when the ratio parallel to mu - mu*parallel to/c is small enough, where c is the penalty parameter that exceeds a threshold c(*) > 0 and mu* is the multiplier corresponding to a local minimizer. Moreover, we prove that the sequence of multiplier vectors generated by the augmented Lagrangian method has at least Q-linear convergence if the sequence of penalty parameters {c(k)} is bounded and the convergence rate is superlinear if {c(k)} is increasing to infinity. Finally, we use a direct way to establish the rate of convergence of the augmented Lagrangian method for the minimax problem with a quadratic objective function and linear equality constraints.