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

arXiv: https://arxiv.org/abs/2301.08345 · HTML: https://mlsanity.com/paper/2301.08345/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 19 Jan 2023.

## Built on by (most cited first, 1 of 1)

- [Outlier Detection Using Generative Models with Theoretical Performance Guarantees](https://mlsanity.com/paper/2310.09999/) (2023)

## Abstract (arXiv)

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 O(√ρ 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.