Mathematics

Questions tagged [fourier-motzkin]

Fourier–Motzkin elimination is an algorithm for eliminating variables from a system of linear inequalities.

13 questions
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Prove Farkas Lemma using the Fourier-Motzkin elimination algorithm

I am trying to prove the Farkas Lemma using the Fourier-Motzkin elimination algorithm. From Wikipedia: Let A be an $m \times n$ matrix and $b$ an $m$-dimensional vector. Then, exactly one of the following two statements is true: There exists an $x…
nikosdi
  • 267
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2 answers

Minimal set of inequalities

I have a set of $m$ linear inequalities in $R^n$, of the form $$ A x \leq b $$ These are automatically generated from the specification of my problem. Many of them could be removed because they are implied by the others. I would like to find the…
knulp
  • 181
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votes
5 answers

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 point in $P$? Is there a specific method, or should I…
HarrySames
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Fourier Motzkin Elimination and totally unimodularity

Suppose $A\in \mathbb R^{m\times n}$ and $b\in \mathbb R^m$, and $A$ is totally unimodular (TUM). For the system $$Ax\leq b,$$ suppose I use Fourier-Motzkin elimination to eliminate first $k$ variables $(x_1,...,x_k)$, $1\leq k<n$ and obtain the new…
user91360
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1 answer

Does Fourier-Motzkin elimination change the solution set?

The Fourier-Motzkin elimination (FME) is a procedure that reduces an $n$ -variable problem to an equivalent $(n − 1)$ -variable problem. It is analogous to Gaussian elimination but for a system of inequalities. In this post, it was shown why…
Adrian
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Fourier Motzkin elimination with positive coefficients only

How can we use Fourier-Motzkin elimination on system of inequalities with positive coefficients preceding each variable $x_1$ to $x_2$ . Obviously, in this case we will only have an upper bound as a solution but how do we find it. For example: $5x_1…
chaosmonk
  • 123
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1 answer

Linear program with two equality constraints

Compute the minimal value of $$x_1 + 2x_2 + 3x_3$$ when $x_1$, $x_2$, $x_3$ satisfy $$x_1 − 2x_2 + x_3 = 4$$ $$−x_1 + 3x_2 = 5$$ and $$x_1 \ge 0, \qquad x_2 \ge 0, \qquad x_3 \ge 0$$ I thought of using Fourier-Motzkin elimination, but we only worked…
user329386
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vote
2 answers

Why is the Farkas Lemma so popular?

When studying optimization, the Farkas Lemma is a very common occurrence. It is used in order to obtain a solvability criterion for linear programs, since it states that exactly one of two problems is solvable. Currently I am doing some research on…
samabu
  • 789
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Solutions to system using Fourier-Motzkin Elimination

I am trying to find a solution to this system using Fourier-Motzkin Elimination, but I don't know how to finish this. Here is what I have so far. $x_1-x_2\leq 0,\quad x_1-x_3\leq 0,\quad -2x_1+2x_2+4x_3\leq 0,\quad -x_3\leq -1$ $\iff\begin{cases}…
user619755
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1 answer

why "$7y\leq 29$, y odd or $7y \leq 26$, y even"?

When reading an example in "Fourier-Motzkin elimination extension to integer programming problems-H.P.Williams" : "Suppose we wished lo eliminate x between the following two inequalities \begin{align*} 3x+5y \leq 19\quad(1)\\ 2x+y \geq 3\quad (2)…
Thu Le
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Fourier-Motzkin Elimination for Single Vertex Computation

Given a convex polytope $P = \{ x \in \mathbb{R}^n \mid Ax \leq b \}$ , I want to know whether we can use Fourier-Motzkin elimination (or an adaptation therefore) to compute one vertex of $P$ (or to determine that $P$ has no vertices). I know that…
Mens
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Find the optimal cost using Fourier-Motzkin elimination

Consider the following linear program $$ \begin{array}{ll} \underset{x_1, x_2} {\text{minimize}} & 50 x_1 + 80 x_2 \\ \text{subject to} & 2 x_1 + 8 x_2 \leq 5 \\ & 6 x_1 + 5 x_2 \leq 10 \\ & x_1, x_2 \geq 0\end{array} $$ Find the optimal cost of…
python-dude153
  • 11
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1 answer

Find Inverse Fourier Transform

Find the inverse Fourier Transform of $$ { F(\omega)=\frac{1}{2\pi(a+j\omega)^2} \ } $$ using the convolution theorem. Hint: the Fourier Transform of $e^{-at} u(t)=\frac{1}{\sqrt{2\pi}(a+j\omega)} $
Ralph
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