# Questions tagged [linear-programming]

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

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### Modeling problem (of Operational Research)

Consider a logistics system consisting of $n$ production sites and $m$ warehouses. For a given product, the monthly production capacity of the production sites is $p_i$ units, with $i = 1,\dots, n$. ...
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### Solving a non-linear constrained linear function optimization

Given $k \in \mathbb{N}$, the $k$-vector-norm is defined as the sum of the $k$ largest entries of a vector (largest w.r.t. to absolute value). So if $k=1$, then the $k$-norm is actually the supremum ...
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### Efficiently solving system of matrix equations

Suppose I have the two matrix equations: $$A_1 M = C_1$$ $$A_2 M = C_2$$ where $A_1, A_2, C_1,C_2$ are given. I want to know if there is a matrix $M$ with entries in $\{0,1\}$ that satisfies the ...
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### Polynomial time for a quadratic equation and linear inequalities?

Does anyone know how to find a feasible solution (or the infeasibility of any solution) in a polynomial time to the following problem: \begin{align*} xAx^t = 0, \\ Bx^t = c, \\ x_i \ge 0, \end{align*} ...
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### Showing that the linear program is unbounded

Let $A\in\Bbb{R}^{m\times n},\ b\in\Bbb{R}^m, c\in\Bbb{R}^n.$ Consider the linear program:$$\;\;\;\;\qquad \max c^Tx\\ \text{s.t.}\quad Ax\leq b$$Assume that there is a feasible $w$, with $c^Tw\gt 0$ ...
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### Multiple variable optimization methods with constraints

This is something I'm doing for a video game so may see some nonsense in the examples I provide. Here's the problem: I want to get a specific amount minerals, to get this minerals I need to refine ore....
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### Why we find out the initial feasible solution for transportation problem?

I saw several methods that available to obtain an initial basic feasible solution of a transportation problem. ...
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### A question about implementation of Farkas lemma

The Farkas Lemma: Let $A$ be an $m\times n$ matrix, $b\in\mathcal{R}^m$. Then exactly one of the following two assertions is true: (1) There exists an $x\in \mathcal{R}^n$ such that $Ax=b$ and $x\ge0$...
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### Linear Programming Basic Solution. Could someone help?

so I am working through the proofs and reading the book "Linear and NonLinear Programming" by Luenberger and wanted to ask for some help. If someone could read the following extract and ...
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### How can model this conditional constraint?

I have a known matrix, $S$ of size $N_U\times N_B$. For each row, the elements are sorted in ascending order. I have also defined a binary variable $X$ of size $N_U\times N_B$. I want to formulate a ...
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### compute primal variables based on dual answer in linear programming

I have an optimization problem (primal problem) which is solved by the duality theorem. So I have constraints of the dual and its variable's value. it is worth mentioning the problem is linear. how ...
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### Can the number of columns be less than the dimension?

I don't see how $k < m$ can possibly occur in part (b). Imagine $X \in R^{n, k}$ describes a matrix for the vectors $x^1$ to $x^K$. $m = \dim( span(S)) \leq \min(n,k)$ Then if $k < m$ in part (b)...