A nonlinear augmented Lagrangian for constrained minimax problems (Q427029)

This paper presents a nonlinear augmented Lagrangian for solving minimax problems with inequality constraints. The corresponding algorithm mainly consists of minimizing the nonlinear augmented Lagrangian function and updating the Lagrange multipliers and controlling parameter.The convergence theory shows that the corresponding algorithm is locally Q-superlinear convergent when the controlling parameter is less than a threshold under the mild conditions.NEWLINENEWLINEMotivated by the importance of numerical stability for the algorithm, this paper studies the condition number of the Hessian of the nonlinear augmented Lagrangian function, in which the condition number is proportional to the reciprocal of the controlling parameter, which means that the controlling parameter cannot be very small when implementing the algorithm and accords with the convergence result of the algorithm. The theoretical results are validated further by the preliminary numerical experiments for some typical minimax problems, which show that the nonlinear augmented Lagrangian is promising.