Fourier–Motzkin elimination is an algorithm for eliminating variables from a system of linear inequalities.
Questions tagged [fourier-motzkin]
12 questions
10
votes
1 answer
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
8
votes
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
5
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…
HarrySames
- 151
4
votes
0 answers
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
user91360
- 1,231
2
votes
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 Gaussian…
Adrian
- 2,076
2
votes
1 answer
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
2
votes
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…
user329386
- 21
- 1
1
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
1
vote
0 answers
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…
user619755
- 620
1
vote
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…
Thu Le
- 433
0
votes
0 answers
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…
0
votes
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
- 11