404, that's not optimal.

\begin{alignat*}{4} \min &\enspace& 100 x_1 & ~- & ~25 x_2 \\ s.t.\ && x_1 &~-& x_2 & ~\geq&& \enspace 1\\ && 3x_1 &~+& x_2 & ~\geq && \enspace16\\ && x_1 &~,& x_2 & ~\in && \enspace\mathbb{Z}_+\\ \end{alignat*}

This is a very simple integer program. You may solve it by examination. If you try it yourself you will find: the value 400 is optimal, not 404. In practice, integer programs do not only have two variables, but hundreds of thousands. Examination will not work anymore since there are incredibly many possibilities.

Our mathematical algorithms can still compute optimal solutions. Provably! Maybe also for your problem?