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In this paper, we consider a nonconvex optimization problem with nonlinearequality constraints. We assume that both, the objective function and thefunctional constraints are locally smooth. For solving this problem, we proposea linearized augmented Lagrangian method, i.e., we linearize the functionalconstraints in the augmented Lagrangian at the current iterate and add aquadratic regularization, yielding a subproblem that is easy to solve, andwhose solution is the next iterate. Under a dynamic regularization parameterchoice, we prove global asymptotic convergence of the iterates to a criticalpoint of the problem. We also derive convergence guarantees for the iterates ofour method to an $\epsilon$ first-order optimal solution in$\mathcal{O}(1/{\epsilon^2})$ outer iterations. Finally, we show that, when theproblem data are e.g., semialgebraic, the sequence generated by our algorithmconverges and we derive convergence rates. We validate the theory and theperformance of the proposed algorithm by numerically comparing it with theexisting methods from the literature.

Complexity of linearized augmented Lagrangian for optimization with nonlinear equality constraints