Find a valid inequality

Question

Find a valid inequality for

$$ \{x\in\{0,1\}^5 \mid 9x_1 + 8x_2 + 6x_3 + 6x_4 + 5x_5 \leq 14\} $$

that cuts off $(1/4, 1/8, 3/4, 3/4, 0)$.

I tried both Chvàtal cut and cover inequality, both of which don’t work.

Answer 1

Of the $32$ vertices of the $5$-cube, $14$ satisfy your constraint.
If $w = (1/4,1/8,3/4,3/4,0)$, try maximizing $y \cdot w - c$ subject to $y \cdot v \le c$ for those vertices $v$, where say $-1 \le y_i \le 1$.