On minimizing a well behaved function with equality and box constraint I'm wondering if it was studied an algorithm which is some sort of a mix of augmented lagrangean and projected gradient, basically you compute the augmented lagrangean of the restriction but without penalizing the box, then the subproblem becomes $$\begin{array}{ll} \min & L(x,\mu,\rho) & \\ s.a & l \leq x \leq u\end{array} $$
Normally one would solve the subproblem and then update the multiplier as $\lambda=\lambda+\frac{\rho}{2}h(x)$ but what I was thinking of is computing $\bar{x}=argmin \nabla_xL(x,\mu,\rho)=0$, then project $\bar{x}$ on the box then update the multiplier every time, so the subproblem isn't actually solved at each iteration.
Does anyone know if this was already done or studied before? I searched and I didn't find anything on it...