Mixing
Linear Programming Negativity Constraints
$begingroup$
What happens when a variable is negative?
An example would be:
Maximize z = 3x1 + 4x2, subject to constraints:
- 2x1 + 3x2 <= 10
- 2x1 - 4x2 <= 20
- x2 <= 10
- x1 >= 0
To set up an Linear Programming problem in Standard Form, I learned that it must be of maximization type. The constraints must be <= (which is good as 1) and 2) agree with that). However, x1 is >= 0, but x2 is not. What would one do in this case? I tried introducing slack variables, namely x2', but I don't know where to go from here, any hints/help would be appreciated. I am just confused on what to do when one of the variables does not satisfy the positivity constraint, here x2<=10.
Edit: I think I can rewrite x2<= 10 as x2-10 <= 0. Then introduce x2' = x2-10. Then, can I replace x2 with x2'+10?
linear-algebra optimization linear-programming
asked Feb 1 at 19:58
Taffies1Taffies1
52
$endgroup$
add a comment |
$begingroup$
What happens when a variable is negative?
An example would be:
Maximize z = 3x1 + 4x2, subject to constraints:
- 2x1 + 3x2 <= 10
- 2x1 - 4x2 <= 20
- x2 <= 10
- x1 >= 0
To set up an Linear Programming problem in Standard Form, I learned that it must be of maximization type. The constraints must be <= (which is good as 1) and 2) agree with that). However, x1 is >= 0, but x2 is not. What would one do in this case? I tried introducing slack variables, namely x2', but I don't know where to go from here, any hints/help would be appreciated. I am just confused on what to do when one of the variables does not satisfy the positivity constraint, here x2<=10.
Edit: I think I can rewrite x2<= 10 as x2-10 <= 0. Then introduce x2' = x2-10. Then, can I replace x2 with x2'+10?
linear-algebra optimization linear-programming
asked Feb 1 at 19:58
Taffies1Taffies1
52
$endgroup$
add a comment |
$begingroup$
What happens when a variable is negative?
An example would be:
Maximize z = 3x1 + 4x2, subject to constraints:
- 2x1 + 3x2 <= 10
- 2x1 - 4x2 <= 20
- x2 <= 10
- x1 >= 0
To set up an Linear Programming problem in Standard Form, I learned that it must be of maximization type. The constraints must be <= (which is good as 1) and 2) agree with that). However, x1 is >= 0, but x2 is not. What would one do in this case? I tried introducing slack variables, namely x2', but I don't know where to go from here, any hints/help would be appreciated. I am just confused on what to do when one of the variables does not satisfy the positivity constraint, here x2<=10.
Edit: I think I can rewrite x2<= 10 as x2-10 <= 0. Then introduce x2' = x2-10. Then, can I replace x2 with x2'+10?
linear-algebra optimization linear-programming
asked Feb 1 at 19:58
Taffies1Taffies1
52
$endgroup$
What happens when a variable is negative?
An example would be:
Maximize z = 3x1 + 4x2, subject to constraints:
- 2x1 + 3x2 <= 10
- 2x1 - 4x2 <= 20
- x2 <= 10
- x1 >= 0
To set up an Linear Programming problem in Standard Form, I learned that it must be of maximization type. The constraints must be <= (which is good as 1) and 2) agree with that). However, x1 is >= 0, but x2 is not. What would one do in this case? I tried introducing slack variables, namely x2', but I don't know where to go from here, any hints/help would be appreciated. I am just confused on what to do when one of the variables does not satisfy the positivity constraint, here x2<=10.
Edit: I think I can rewrite x2<= 10 as x2-10 <= 0. Then introduce x2' = x2-10. Then, can I replace x2 with x2'+10?
linear-algebra optimization linear-programming
linear-algebra optimization linear-programming
asked Feb 1 at 19:58
Taffies1Taffies1
52
asked Feb 1 at 19:58
Taffies1Taffies1
52
edited Feb 1 at 20:14
Taffies1
asked Feb 1 at 19:58
Taffies1Taffies1
52
asked Feb 1 at 19:58
Taffies1Taffies1
52
asked Feb 1 at 19:58
Taffies1Taffies1
52
52
add a comment |
add a comment |
1 Answer
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$begingroup$
Replace $x_2$ with $x_2^+ - x_2^-$, with $x_2^+ geq 0$ and $x_2^- geq 0$.
answered Feb 1 at 22:34
LinAlgLinAlg
10.1k1521
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add a comment |
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$begingroup$
Replace $x_2$ with $x_2^+ - x_2^-$, with $x_2^+ geq 0$ and $x_2^- geq 0$.
answered Feb 1 at 22:34
LinAlgLinAlg
10.1k1521
$endgroup$
add a comment |
$begingroup$
Replace $x_2$ with $x_2^+ - x_2^-$, with $x_2^+ geq 0$ and $x_2^- geq 0$.
answered Feb 1 at 22:34
LinAlgLinAlg
10.1k1521
$endgroup$
add a comment |
$begingroup$
Replace $x_2$ with $x_2^+ - x_2^-$, with $x_2^+ geq 0$ and $x_2^- geq 0$.
answered Feb 1 at 22:34
LinAlgLinAlg
10.1k1521
$endgroup$
Replace $x_2$ with $x_2^+ - x_2^-$, with $x_2^+ geq 0$ and $x_2^- geq 0$.
answered Feb 1 at 22:34
LinAlgLinAlg
10.1k1521
answered Feb 1 at 22:34
LinAlgLinAlg
10.1k1521
answered Feb 1 at 22:34
LinAlgLinAlg
10.1k1521
answered Feb 1 at 22:34
LinAlgLinAlg
10.1k1521
10.1k1521
add a comment |
add a comment |
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