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

TL;DR

This paper addresses nonconvex optimization with nonlinear equality constraints by introducing the Linearized Augmented Lagrangian (L-AL) method, which applies a Gauss-Newton style linearization of both the objective and constraints inside the augmented Lagrangian and adds a dynamic quadratic regularization to yield a simple quadratic subproblem at each iteration. The dual update is also linearized, enabling a robust convergence analysis that proves global convergence to a KKT point under LICQ and the KL property, and establishes an $\epsilon$-first-order complexity of $O(\sqrt{\rho}\epsilon^{-2})$ Jacobian evaluations, with improved second-order guarantees under benign nonconvexity. The work combines the strengths of Proximal AL and SCP approaches, offering global guarantees, avoidance of heavy subroutines, and efficiency for large-scale problems, with numerical experiments comparing against SCP, IPOPT, and Algencan. It also discusses a practical inner-loop strategy to adaptively choose the penalty parameter $\rho$ when problem parameters are unknown, enhancing applicability. Overall, the proposed framework advances the theoretical and practical performance of augmented-Lagrangian methods for smooth nonconvex constrained optimization.

Abstract

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 $ε$-first-order solution in $\mathcal{O}(\sqrtρ ε^{-2})$ Jacobian evaluations, where $ρ$ is the penalty parameter. Moreover, when the problem exhibits a benign nonconvex property, we derive improved convergence results to an $ε$-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.