Augmented Lagrangian Method can be used with inequality constraints. The question is how.
One approach (according to Numerical Optimization Book by Nocedal and Wright; page 522), is linearly constrained Lagrangian. Description is shown in the attached image. The question is what $A_k$ in the formulation 17.55b? I do not find any description for that:
The matrix $A_k$ should be an approximation of the Jacobian of $c$ at $x_k$. For the first look, you can think of $\nabla c(x_k)^\top$. Then, the constraint is just the first-order Taylor expansion of $c$ at $x_k$.