Use max operation in a constraint in Linear Programming
Introduce variables $m_1,m_2,m_3$ to represent the three maxes.

Add the linear inequality $m_1 + m_2 + m_3 \ge q$.

Then, add the following extra inequalities for $m_1$:


$0 \le m_1 \le 1$
$m_1 \le x_1 + x_2 + \dots + x_n$
$x_1 \le m_1$, $x_2 \le m_1$, $\dots$, $x_n \le m_1$


and similarly for $m_2$ and $m_3$.