I have a binary variable in an minimizing optimization problem. The algorithm I'm about to use to solve the problem requires the problem to be continues and convex. However, I can use an indicator function for the binary variable and then add it to objective function. so if $z\in\{0,1\}$, we have $g(z)$ as indicator function of binary variable so that if $z\in\{0,1\}$ then $g(z)=0$ and otherwise, $g(z)=\infty$.
in the iterative algorithm that I'm using (ADMM), there is a $z_\text{update}$ step as below: $$z_{\text{update}}=\operatorname{arg\,min}(g(z)+Nρ/2 ‖z-v‖_2^2 )\text{ with respect to }z\in\mathbb R^m$$ Now, the question is, can we express this update step as proximal mapping of $g(z)$ and calculate the minimum value of expression with respect to $z$ as following picture?
