Given $c\in\Bbb R^{n_1},d\in\Bbb R^{n_2}$, $E\in\Bbb R^{n_1\times n_2}$, $A\in\Bbb R^{m_1\times n_1}$, $B\in\Bbb R^{m_2\times n_2}$ $a\in\Bbb R^{m_1}$, $b\in\Bbb R^{m_2}$ and $t\in\Bbb R$ we know finding if there are $x\in\Bbb R^{n_1}$ and $y\in\Bbb R^{n_2}$ such that $$c'x+d'y+x'Ey\leq t$$ $$Ax\leq a$$ $$By\leq b$$ holds is $NP$-hard.
Are there non-trivial scenarios where this problem is in $P$?
An example of a trivial scenarios is when $n_1n_2=0$ as we reduce to linear programming or $E$ is $0$ matrix.
An example of non-trivial scenarios would be special structure on $E,A,B$ and conditions on $c,d$.
