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This case is still NP-hard. Suppose we have an instance of 3-SAT: $F=C_1\wedge\ldots\wedge C_n; C_i=L_{i,1}\vee L_{i,2}\vee L_{i,3}$, where each literal $L_{i,j}$ is either $V$ or $\neg V$ for some …

The conjunction of the first two clauses, $(\sigma_1\cup\sigma_2\cup u_1)(\sigma_3\cup\ldots\cup\sigma_m\cup\bar{u}_1)$ is equisatisfiable to the original clause, as can be easily checked (any valuati …

Without any further constraints, this expression will in general be unbounded, so the maximum won't exist. Let $V$ be $\{v_1,\ldots,v_n\}$ with $n\ge 2$. Pick $i\neq j$ such that $v_i,v_j$ are not bo …

Assuming your polytopes are closed, I don't think this is true. Each of your polytopes $P_1,P_2,Q$ is a product of (possibly unbounded) intervals $I_i^1$, $I_i^2$, or $I_i^Q$, respectively, for $i=1,\ …

The problem as stated now is solvable in linear time. To see this, suppose $p\in P$ is such that there are $x\in X$ and $w\in W$ with $p_i=x_iw_i$ for all $i$. This means on the one hand that $1=\sum …