How to minimize the maximum of $n$ multivariate functions?

Question

If I have $n$ functions $f_i:\mathbb{N}\times \mathbb{N}\rightarrow \mathbb{N}$. What approach should I take to minimize $max(f_i)$?

For example, Consider:

$$f_1(l_1,l_2)=l_1+l_2$$

$$f_2(l_1, l_2)=x_2-l_2+l_1$$

$$f_3(l_1, l_2)=x_1-l_1+l_2$$

$$f_4(l_1, l_2)=x_1+x_2+l_1+l_2$$

for some constants $x_1, x_2$. I want to minimize $max((f_1,f_2,f_3, f_4)(l_1,l_2))$.

Answer 1

You can try to solve the following problem: $$ \underset{l_1\in \mathbb{N},l_2\in \mathbb{N},t}{\text{mininize}}\quad t\\ \text{subject to} \quad f_1(l_1,l_2)\leq t\\ \quad \quad \quad \quad f_2(l_1,l_2)\leq t\\ \quad \quad \quad \quad f_3(l_1,l_2)\leq t\\ \quad \quad \quad \quad f_4(l_1,l_2)\leq t $$