Complexity of a linearized augmented Lagrangian method for nonconvex minimization with nonlinear equality constraints

In this paper, we consider a nonconvex optimization problem with nonlinear equality constraints. We assume that both, the objective function and the functional constraints are locally smooth. For solving this problem, we propose a linearized augmented Lagrangian method, i.e., we linearize the objective function and the functional constraints in a Gauss-Newton fashion at the current iterate within the augmented Lagrangian function and add a quadratic regularization, yielding a subproblem that is easy to solve, and whose solution is the next primal iterate. The update of the dual multipliers is also based on the linearization of functional constraints. Under a novel dynamic regularization parameter choice, we prove boundedness and global asymptotic convergence of the iterates to a first-order solution of the problem. We also derive convergence guarantees for the iterates of our method to an $\epsilon$-first-order solution in $\mathcal{O}(\sqrt{\rho} \epsilon^{-2})$ Jacobian evaluations, where $\rho$ is the penalty parameter. Moreover, when the problem exhibits a benign nonconvex property, we derive improved convergence results to an $\epsilon$-second-order solution. Finally, we validate the performance of the proposed algorithm by numerically comparing it with the existing methods and software from the literature.