I am having some difficulties with optimization problems with inequality constraints. In general the problems I am given will look something like this:
$$\min f(x,y,z) \\ \text{s.t.} \space \space \space g(x,y,z) \le0 \\ \space \space \space \space h(x,y,z)=0$$
I would usually start by writing down the lagrange function $\mathcal L$, where $\lambda$, $\mu$ are the corresponding Lagrange multipliers.
$$\mathcal L(x,y,z,\lambda,\mu)=f(x,y,z)+\lambda (h(x,y,z))+\mu(g(x,y,z))$$
In order to find the KKT-points I would solve:
$$\nabla \mathcal L=0$$
which will give me one or many KKT points. Here is the problem I am having:
This method seems like I am always treating the inequality as an equality. Do I ever treat the "strictly less" case?
Sometimes I see people (lectures or youtube videos I have watched) "ignore" the inequality constraint, solve the problem and then checking if the inequality is satisfied at the resulting points. Why does one do that?
If I am getting a negative lagrange multiplier for the inequality constraint, how do I proceed? Does it just mean there are no points that satisfy that constraint?
Is there some sort of "road map" or strategy when dealing with problems like this (optimization with one inequality and eqality constraint)?