How to assure that we still are in feasible set during optimization iterations?

Question

What are methods to force the point remains in the optimization domain ,other than taking projection step after each iteration?I haven't find any other solution for this but this projection step seems to slow down the algorithm.The problem is minimizing $f(x)+g(z)$(using ADMM method) such that $Ax+Bz+b=0$ and $g$ is the indicator function of a set $c$,and I mean that in iterations some time $z$ is not in $c$ as a result $g(z)$ becomes infinity and the algorithm give back $z=NaN$, so I need to turn back into c after each iteration.any hint would be appreciated.

Answer 1

When $g(z)$ is the indicator function of a convex set $C$, then in the associated minimization step of the ADMM method, you'll be finding a $z$ in $C$ that also minimizes the augmented Lagrangian term

$\min_{z \in C} \frac{\rho}{2} \| Bz - (Ax+b) \|_{2}^{2}$

This is the minimization of a quadratic function over $C$. Your procedure for doing this must return a $z$ in $C$. There aren't any other steps of the algorithm that would change $z$.