I want to find a minimum threshold value for a constraint, such that if this constraint is satisfied, the next one must be satisfied.
For example, given two inequations $f_1(X)\geq a$ and $f_2(X) \geq b$, where $X$ is a vector, and for each $x_i \in X$, $x_i = \{0,1\}$, I want to find a minimum $a$, such that, if $f_1(X)\geq a$ is satisfied, then $f_2(X) \geq b$ must be satisfed for all valied $X$.
Let $\epsilon>0$ be a small tolerance. You want to enforce $$\forall X\in\{0,1\}^k\left(f_2(X)<b \implies f_1(X)<a\right)$$ So take $$a=\epsilon+\max_{X\in\{0,1\}^k: f_2(X)<b} f_1(X)$$ If $f_1$ is integer-valued, you can take $\epsilon=1$.