Absolute value in Linear programming problem
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I have this problem, where I have to reformulate it as a linear programming problem.
Minimize $ 2x_1 +3 vert x_2-10vert $
subject to $vert x_1+2vert + vert x_2vert ≤5 $
The subject is new to me. Can anyone help?
linear-programming
asked Sep 6 at 9:45
Gabarta123
293
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up vote
0
down vote
favorite
I have this problem, where I have to reformulate it as a linear programming problem.
Minimize $ 2x_1 +3 vert x_2-10vert $
subject to $vert x_1+2vert + vert x_2vert ≤5 $
The subject is new to me. Can anyone help?
linear-programming
asked Sep 6 at 9:45
Gabarta123
293
$tgeq |x|$ is equivalent to $tgeq x$ and $tgeq -x$.
– Michal Adamaszek
Sep 6 at 10:01
1
Don´t forget to mark answers as accepted.
– callculus
Sep 6 at 13:46
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I have this problem, where I have to reformulate it as a linear programming problem.
Minimize $ 2x_1 +3 vert x_2-10vert $
subject to $vert x_1+2vert + vert x_2vert ≤5 $
The subject is new to me. Can anyone help?
linear-programming
asked Sep 6 at 9:45
Gabarta123
293
I have this problem, where I have to reformulate it as a linear programming problem.
Minimize $ 2x_1 +3 vert x_2-10vert $
subject to $vert x_1+2vert + vert x_2vert ≤5 $
The subject is new to me. Can anyone help?
linear-programming
linear-programming
asked Sep 6 at 9:45
Gabarta123
293
asked Sep 6 at 9:45
Gabarta123
293
asked Sep 6 at 9:45
Gabarta123
293
asked Sep 6 at 9:45
Gabarta123
293
asked Sep 6 at 9:45
Gabarta123
293
293
$tgeq |x|$ is equivalent to $tgeq x$ and $tgeq -x$.
– Michal Adamaszek
Sep 6 at 10:01
1
Don´t forget to mark answers as accepted.
– callculus
Sep 6 at 13:46
add a comment |Â
$tgeq |x|$ is equivalent to $tgeq x$ and $tgeq -x$.
– Michal Adamaszek
Sep 6 at 10:01
1
Don´t forget to mark answers as accepted.
– callculus
Sep 6 at 13:46
$tgeq |x|$ is equivalent to $tgeq x$ and $tgeq -x$.
– Michal Adamaszek
Sep 6 at 10:01
$tgeq |x|$ is equivalent to $tgeq x$ and $tgeq -x$.
– Michal Adamaszek
Sep 6 at 10:01
1
1
Don´t forget to mark answers as accepted.
– callculus
Sep 6 at 13:46
Don´t forget to mark answers as accepted.
– callculus
Sep 6 at 13:46
add a comment |Â
2 Answers
2
active
oldest
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up vote
1
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Alternatively, the constraint $vert x_1+2vert + vert x_2vert ≤5$ corresponds to:
$$pm(x_1+2)pm x_2 le 5 iff
begincases x_1+2+x_2le 5\
-(x_1+2)+x_2le 5\
x_1+2-x_2le 5\
-(x_1+2)-x_2le 5\
endcases$$
The feasible region is:
$hspace3cm$
Since $x_2<10$, then the original problem can be reformulated as:
$$textMinimize 2x_1+3(10-x_2) textsubject to:\
begincasesx_1+x_2le 3\
x_1-x_2ge -7\
x_1-x_2le 3\
x_1+x_2ge -7\
endcases$$
The answer is: $f(-2,5)=11$. To be verified: Original and Reformulated.
answered Sep 6 at 20:07
farruhota
15.4k2734
add a comment |Â
up vote
-1
down vote
Minimize $2x_1 +3 vert x_2-10vert$ subject to $vert x_1+2vert + vert x_2vert ≤5$.
It can be converted to $6$ linear programming problems:
$$1) begincases2x_1-3(x_2-10) to textmin\ -(x_1+2) -(x_2)≤5\
colorredx_1le -2\
x_2le0endcases
4) begincases2x_1-3(x_2-10) to textmin\ (x_1+2) -(x_2)≤5\
colorbluex_1ge -2\
x_2le0endcases \
2) begincases2x_1-3(x_2-10) to textmin\ -(x_1+2) +(x_2)≤5\
colorredx_1le -2 \
0le x_2le10 endcases
5) begincases2x_1-3(x_2-10) to textmin\ (x_1+2) +(x_2)≤5\
colorbluex_1ge -2\
0le x_2le10endcases\\
3) begincases2x_1+3(x_2-10) to textmin\ -(x_1+2) +(x_2)≤5\
colorredx_1le -2\
10le x_2 endcases
6) begincases2x_1+3(x_2-10) to textmin\ (x_1+2) +(x_2)≤5\
colorbluex_1ge -2\
10le x_2 endcases\$$
answered Sep 6 at 13:24
farruhota
15.4k2734
Although mathematically correct, your answer is highly inefficient. You can solve it as one linear optimization problem. See the comment of Michal Adamaszek.
– LinAlg
Sep 6 at 17:39
@LinAlg, thanks for the comment, added another answer. Left this one for reference, since it is a straightforward (though longer) method.
– farruhota
Sep 6 at 20:09
add a comment |Â
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
Alternatively, the constraint $vert x_1+2vert + vert x_2vert ≤5$ corresponds to:
$$pm(x_1+2)pm x_2 le 5 iff
begincases x_1+2+x_2le 5\
-(x_1+2)+x_2le 5\
x_1+2-x_2le 5\
-(x_1+2)-x_2le 5\
endcases$$
The feasible region is:
$hspace3cm$
Since $x_2<10$, then the original problem can be reformulated as:
$$textMinimize 2x_1+3(10-x_2) textsubject to:\
begincasesx_1+x_2le 3\
x_1-x_2ge -7\
x_1-x_2le 3\
x_1+x_2ge -7\
endcases$$
The answer is: $f(-2,5)=11$. To be verified: Original and Reformulated.
answered Sep 6 at 20:07
farruhota
15.4k2734
add a comment |Â
up vote
1
down vote
Alternatively, the constraint $vert x_1+2vert + vert x_2vert ≤5$ corresponds to:
$$pm(x_1+2)pm x_2 le 5 iff
begincases x_1+2+x_2le 5\
-(x_1+2)+x_2le 5\
x_1+2-x_2le 5\
-(x_1+2)-x_2le 5\
endcases$$
The feasible region is:
$hspace3cm$
Since $x_2<10$, then the original problem can be reformulated as:
$$textMinimize 2x_1+3(10-x_2) textsubject to:\
begincasesx_1+x_2le 3\
x_1-x_2ge -7\
x_1-x_2le 3\
x_1+x_2ge -7\
endcases$$
The answer is: $f(-2,5)=11$. To be verified: Original and Reformulated.
answered Sep 6 at 20:07
farruhota
15.4k2734
add a comment |Â
up vote
1
down vote
up vote
1
down vote
Alternatively, the constraint $vert x_1+2vert + vert x_2vert ≤5$ corresponds to:
$$pm(x_1+2)pm x_2 le 5 iff
begincases x_1+2+x_2le 5\
-(x_1+2)+x_2le 5\
x_1+2-x_2le 5\
-(x_1+2)-x_2le 5\
endcases$$
The feasible region is:
$hspace3cm$
Since $x_2<10$, then the original problem can be reformulated as:
$$textMinimize 2x_1+3(10-x_2) textsubject to:\
begincasesx_1+x_2le 3\
x_1-x_2ge -7\
x_1-x_2le 3\
x_1+x_2ge -7\
endcases$$
The answer is: $f(-2,5)=11$. To be verified: Original and Reformulated.
answered Sep 6 at 20:07
farruhota
15.4k2734
Alternatively, the constraint $vert x_1+2vert + vert x_2vert ≤5$ corresponds to:
$$pm(x_1+2)pm x_2 le 5 iff
begincases x_1+2+x_2le 5\
-(x_1+2)+x_2le 5\
x_1+2-x_2le 5\
-(x_1+2)-x_2le 5\
endcases$$
The feasible region is:
$hspace3cm$
Since $x_2<10$, then the original problem can be reformulated as:
$$textMinimize 2x_1+3(10-x_2) textsubject to:\
begincasesx_1+x_2le 3\
x_1-x_2ge -7\
x_1-x_2le 3\
x_1+x_2ge -7\
endcases$$
The answer is: $f(-2,5)=11$. To be verified: Original and Reformulated.
answered Sep 6 at 20:07
farruhota
15.4k2734
answered Sep 6 at 20:07
farruhota
15.4k2734
answered Sep 6 at 20:07
farruhota
15.4k2734
answered Sep 6 at 20:07
farruhota
15.4k2734
15.4k2734
add a comment |Â
add a comment |Â
up vote
-1
down vote
Minimize $2x_1 +3 vert x_2-10vert$ subject to $vert x_1+2vert + vert x_2vert ≤5$.
It can be converted to $6$ linear programming problems:
$$1) begincases2x_1-3(x_2-10) to textmin\ -(x_1+2) -(x_2)≤5\
colorredx_1le -2\
x_2le0endcases
4) begincases2x_1-3(x_2-10) to textmin\ (x_1+2) -(x_2)≤5\
colorbluex_1ge -2\
x_2le0endcases \
2) begincases2x_1-3(x_2-10) to textmin\ -(x_1+2) +(x_2)≤5\
colorredx_1le -2 \
0le x_2le10 endcases
5) begincases2x_1-3(x_2-10) to textmin\ (x_1+2) +(x_2)≤5\
colorbluex_1ge -2\
0le x_2le10endcases\\
3) begincases2x_1+3(x_2-10) to textmin\ -(x_1+2) +(x_2)≤5\
colorredx_1le -2\
10le x_2 endcases
6) begincases2x_1+3(x_2-10) to textmin\ (x_1+2) +(x_2)≤5\
colorbluex_1ge -2\
10le x_2 endcases\$$
answered Sep 6 at 13:24
farruhota
15.4k2734
Although mathematically correct, your answer is highly inefficient. You can solve it as one linear optimization problem. See the comment of Michal Adamaszek.
– LinAlg
Sep 6 at 17:39
@LinAlg, thanks for the comment, added another answer. Left this one for reference, since it is a straightforward (though longer) method.
– farruhota
Sep 6 at 20:09
add a comment |Â
up vote
-1
down vote
Minimize $2x_1 +3 vert x_2-10vert$ subject to $vert x_1+2vert + vert x_2vert ≤5$.
It can be converted to $6$ linear programming problems:
$$1) begincases2x_1-3(x_2-10) to textmin\ -(x_1+2) -(x_2)≤5\
colorredx_1le -2\
x_2le0endcases
4) begincases2x_1-3(x_2-10) to textmin\ (x_1+2) -(x_2)≤5\
colorbluex_1ge -2\
x_2le0endcases \
2) begincases2x_1-3(x_2-10) to textmin\ -(x_1+2) +(x_2)≤5\
colorredx_1le -2 \
0le x_2le10 endcases
5) begincases2x_1-3(x_2-10) to textmin\ (x_1+2) +(x_2)≤5\
colorbluex_1ge -2\
0le x_2le10endcases\\
3) begincases2x_1+3(x_2-10) to textmin\ -(x_1+2) +(x_2)≤5\
colorredx_1le -2\
10le x_2 endcases
6) begincases2x_1+3(x_2-10) to textmin\ (x_1+2) +(x_2)≤5\
colorbluex_1ge -2\
10le x_2 endcases\$$
answered Sep 6 at 13:24
farruhota
15.4k2734
Although mathematically correct, your answer is highly inefficient. You can solve it as one linear optimization problem. See the comment of Michal Adamaszek.
– LinAlg
Sep 6 at 17:39
@LinAlg, thanks for the comment, added another answer. Left this one for reference, since it is a straightforward (though longer) method.
– farruhota
Sep 6 at 20:09
add a comment |Â
up vote
-1
down vote
up vote
-1
down vote
Minimize $2x_1 +3 vert x_2-10vert$ subject to $vert x_1+2vert + vert x_2vert ≤5$.
It can be converted to $6$ linear programming problems:
$$1) begincases2x_1-3(x_2-10) to textmin\ -(x_1+2) -(x_2)≤5\
colorredx_1le -2\
x_2le0endcases
4) begincases2x_1-3(x_2-10) to textmin\ (x_1+2) -(x_2)≤5\
colorbluex_1ge -2\
x_2le0endcases \
2) begincases2x_1-3(x_2-10) to textmin\ -(x_1+2) +(x_2)≤5\
colorredx_1le -2 \
0le x_2le10 endcases
5) begincases2x_1-3(x_2-10) to textmin\ (x_1+2) +(x_2)≤5\
colorbluex_1ge -2\
0le x_2le10endcases\\
3) begincases2x_1+3(x_2-10) to textmin\ -(x_1+2) +(x_2)≤5\
colorredx_1le -2\
10le x_2 endcases
6) begincases2x_1+3(x_2-10) to textmin\ (x_1+2) +(x_2)≤5\
colorbluex_1ge -2\
10le x_2 endcases\$$
answered Sep 6 at 13:24
farruhota
15.4k2734
Minimize $2x_1 +3 vert x_2-10vert$ subject to $vert x_1+2vert + vert x_2vert ≤5$.
It can be converted to $6$ linear programming problems:
$$1) begincases2x_1-3(x_2-10) to textmin\ -(x_1+2) -(x_2)≤5\
colorredx_1le -2\
x_2le0endcases
4) begincases2x_1-3(x_2-10) to textmin\ (x_1+2) -(x_2)≤5\
colorbluex_1ge -2\
x_2le0endcases \
2) begincases2x_1-3(x_2-10) to textmin\ -(x_1+2) +(x_2)≤5\
colorredx_1le -2 \
0le x_2le10 endcases
5) begincases2x_1-3(x_2-10) to textmin\ (x_1+2) +(x_2)≤5\
colorbluex_1ge -2\
0le x_2le10endcases\\
3) begincases2x_1+3(x_2-10) to textmin\ -(x_1+2) +(x_2)≤5\
colorredx_1le -2\
10le x_2 endcases
6) begincases2x_1+3(x_2-10) to textmin\ (x_1+2) +(x_2)≤5\
colorbluex_1ge -2\
10le x_2 endcases\$$
answered Sep 6 at 13:24
farruhota
15.4k2734
answered Sep 6 at 13:24
farruhota
15.4k2734
answered Sep 6 at 13:24
farruhota
15.4k2734
answered Sep 6 at 13:24
farruhota
15.4k2734
15.4k2734
Although mathematically correct, your answer is highly inefficient. You can solve it as one linear optimization problem. See the comment of Michal Adamaszek.
– LinAlg
Sep 6 at 17:39
@LinAlg, thanks for the comment, added another answer. Left this one for reference, since it is a straightforward (though longer) method.
– farruhota
Sep 6 at 20:09
add a comment |Â
Although mathematically correct, your answer is highly inefficient. You can solve it as one linear optimization problem. See the comment of Michal Adamaszek.
– LinAlg
Sep 6 at 17:39
@LinAlg, thanks for the comment, added another answer. Left this one for reference, since it is a straightforward (though longer) method.
– farruhota
Sep 6 at 20:09
Although mathematically correct, your answer is highly inefficient. You can solve it as one linear optimization problem. See the comment of Michal Adamaszek.
– LinAlg
Sep 6 at 17:39
Although mathematically correct, your answer is highly inefficient. You can solve it as one linear optimization problem. See the comment of Michal Adamaszek.
– LinAlg
Sep 6 at 17:39
@LinAlg, thanks for the comment, added another answer. Left this one for reference, since it is a straightforward (though longer) method.
– farruhota
Sep 6 at 20:09
@LinAlg, thanks for the comment, added another answer. Left this one for reference, since it is a straightforward (though longer) method.
– farruhota
Sep 6 at 20:09
add a comment |Â
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$tgeq |x|$ is equivalent to $tgeq x$ and $tgeq -x$.
– Michal Adamaszek
Sep 6 at 10:01
1
Don´t forget to mark answers as accepted.
– callculus
Sep 6 at 13:46