I am wondering if there is any single step analysis for the Augmented Lagrangian method. Specifically, the problem is $$\min f(x) \text { s.t. } A x=b$$ where $f$ is convex, smooth. Such an objective with linear constraint may be the easiest to deal with. When we do the update, we solve $$\min _{x} f(x)+\overline{\lambda}^{T}(A x-b)+\frac{\rho}{2}\|A x-b\|^{2}=\mathcal{L}(x, \overline{\lambda} ; \rho)$$ by doing gradient update on this augmented Lagrangian and then update dual by $$\lambda=\overline{\lambda}+\rho(A x-b)$$

By saying "single step analysis", I want to bound the one-step progress $f\left(x^{k+1}\right)-f\left(x^{k}\right)$. Such a bound is easily derived for gradient descent, assuming that $f$ is continuous differentiable and has Lipschitz gradient $$\begin{aligned} f\left(x^{k+1}\right)-f\left(x^{k}\right) \leq\left\langle x^{k+1}-x^{k}, \nabla f\left(x^{k}\right)\right\rangle+\frac{L}{2}\left\|x^{k+1}-x^{k}\right\|^{2} \\=\left\langle-\alpha \nabla f\left(x^{k}\right),\right.& \nabla f\left(x^{k}\right) \rangle+\frac{L}{2} \alpha^{2}\left\|\nabla f\left(x^{k}\right)\right\|^{2} \\ &=-\alpha\left(1-\frac{L \alpha}{2}\right)\left\|\nabla f\left(x^{k}\right)\right\|^{2} \end{aligned}$$ This bound is in terms of $\left\|\nabla f\left(x^{k}\right)\right\|^{2} $. But I cannot find a similar one for Augmented Lagrangian method. You can make additional assumption, if necessary. Can anyone have a reference, or a way to derive that? Many thanks!