This paper focuses on the augmented Lagrangian method for solving minimax optimization problems with inequality constraints. The rate of convergence of the augmented Lagrange method is analyzed under a set of sufficiency optimality conditions for the minimax problem. For a given multiplier vector \((\mu ,\lambda )\) , it is demonstrated that the rate of convergence of the augmented Lagrangian method is linear with respect to \(\Vert (\mu ,\lambda )-(\mu ^*,\lambda ^*)\Vert \) and the ratio constant is proportional to 1/c when the ratio \(\Vert (\mu ,\lambda )-(\mu ^*,\lambda ^*)\Vert /c\) is small enough, where c is the penalty parameter that exceeds a threshold \(c_*> 0\) and \((\mu ^*,\lambda ^*)\) is the multiplier corresponding to a local minimax point. Moreover, by analyzing the second-order derivative of the value function of the inequality constrained minimax optimization problem, 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.
