# Tagged Questions

Questions on linear programming, the optimization of a linear function subject to linear constraints.

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### Feasible point of a system of linear inequalities

Let $P$ denote $(x,y,z)\in \mathbb R^3$, which satisfies the inequalities: $$-2x+y+z\leq 4$$ $$x \geq 1$$ $$y\geq2$$ $$z \geq 3$$ $$x-2y+z \leq 1$$ $$2x+2y-z \leq 5$$ How do I find an interior ...
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### Minimizing a function including max functions

Consider the following problem. Let $\mathcal{N} = \{1,2,\ldots,N\}$ and $\mathcal{N}^i = \mathcal{N}\setminus \{i\}$. For each $i \in \mathcal{N}$ and for each $S \subset \mathcal{N}^i$, we have a ...
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### Ray between vertex and point inside a polytope

Let $P$ be a polytope, let $v \in P$ be a vertex of $P$, and let $x \in P$ such that $x$ is not a vertex. Consider the ray $$\forall t>0, \phi(t)=v+t(x-v).$$ Let $t_0$ be the maximal $t$ such that ...
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### Show that exactly one of the following two systems has a solution.

Let A be a $m \times n$ matrix, $\mathbf{c}$ an $n$-dimensional ector and $\mathbf{b} \ge \mathbf{0}$ an $m$-dimensional vector. Show that exactly one of the following two systems has a solution: \$\...