Is it possible to change the constraint in the optimization problem in following way?

Question

I want to solve an optimization problem. One of the constraint is given as follows $$\max(0,g(x))\geq c_1$$ where $c_1\geq 0$ and $g(x)$ is a concave function which can also have negative values. For any $c_1>0$ can I change the above constraint to the following constraint? $$g(x)\geq c_1$$ I will be very thankful for your help in this regard.

Answer 1

For a particular $x$, if $g(x)<0$, then $\max(0, g(x)) = 0$ and hence $0\ge c_1$ doesn't hold. Hence if $g(x)<0$, then $x$ is not feasible.

Hence for $x$ to be feasible, we need $g(x) \geq 0$, that is $\max(0, g(x)) = g(x).$

Hence, yes, you can change the constraint to $g(x) \ge c_1$.