Finding the Dual of an LP, where variables are not well isolated.
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Could someone guide me through how I could go about finding the dual of the following LP. Let $A subset Bbb R^{m x n}$, and $x = (x_1, ..., x_m)$.
$$text{max } z
\
text{st } z le x^TA_{.j} , forall j in {1, ldots, m}
\
x_1 + ldots + x_m = 1
\
x ge vec{0}
$$
, where $A_{.j}$ represents the j'th column of A.
My confusion is coming from the fact that in this case, $A$ is a constant matrix, but all the $x_i$'s are variables; as is z.
So I can't see how I can apply the standard primal-dual conversion.
Thanks!
optimization linear-programming duality-theorems
asked Nov 19 at 22:52
Bojack Horseman
1117
add a comment |
up vote
0
down vote
favorite
Could someone guide me through how I could go about finding the dual of the following LP. Let $A subset Bbb R^{m x n}$, and $x = (x_1, ..., x_m)$.
$$text{max } z
\
text{st } z le x^TA_{.j} , forall j in {1, ldots, m}
\
x_1 + ldots + x_m = 1
\
x ge vec{0}
$$
, where $A_{.j}$ represents the j'th column of A.
My confusion is coming from the fact that in this case, $A$ is a constant matrix, but all the $x_i$'s are variables; as is z.
So I can't see how I can apply the standard primal-dual conversion.
Thanks!
optimization linear-programming duality-theorems
asked Nov 19 at 22:52
Bojack Horseman
1117
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Could someone guide me through how I could go about finding the dual of the following LP. Let $A subset Bbb R^{m x n}$, and $x = (x_1, ..., x_m)$.
$$text{max } z
\
text{st } z le x^TA_{.j} , forall j in {1, ldots, m}
\
x_1 + ldots + x_m = 1
\
x ge vec{0}
$$
, where $A_{.j}$ represents the j'th column of A.
My confusion is coming from the fact that in this case, $A$ is a constant matrix, but all the $x_i$'s are variables; as is z.
So I can't see how I can apply the standard primal-dual conversion.
Thanks!
optimization linear-programming duality-theorems
asked Nov 19 at 22:52
Bojack Horseman
1117
Could someone guide me through how I could go about finding the dual of the following LP. Let $A subset Bbb R^{m x n}$, and $x = (x_1, ..., x_m)$.
$$text{max } z
\
text{st } z le x^TA_{.j} , forall j in {1, ldots, m}
\
x_1 + ldots + x_m = 1
\
x ge vec{0}
$$
, where $A_{.j}$ represents the j'th column of A.
My confusion is coming from the fact that in this case, $A$ is a constant matrix, but all the $x_i$'s are variables; as is z.
So I can't see how I can apply the standard primal-dual conversion.
Thanks!
optimization linear-programming duality-theorems
optimization linear-programming duality-theorems
asked Nov 19 at 22:52
Bojack Horseman
1117
asked Nov 19 at 22:52
Bojack Horseman
1117
asked Nov 19 at 22:52
Bojack Horseman
1117
asked Nov 19 at 22:52
Bojack Horseman
1117
asked Nov 19 at 22:52
Bojack Horseman
1117
1117
add a comment |
add a comment |
1 Answer
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You could define $x'=begin{pmatrix}x \ zend{pmatrix}$, then the problem is:
$$begin{align}
max quad & begin{pmatrix}0 \ 0 \ vdots \ 0 \ 1end{pmatrix}^T x' \
text{s.t.} quad & begin{pmatrix} A^T & -e \ e^T & 0 \ -e^T & 0end{pmatrix}x' leq begin{pmatrix} 0 \ 0 \ vdots \ 0 \ 1 \ -1end{pmatrix} \
& x' geq 0.
end{align}$$
Deriving the dual is now simple.
answered Nov 20 at 0:36
LinAlg
7,9241520
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
accepted
You could define $x'=begin{pmatrix}x \ zend{pmatrix}$, then the problem is:
$$begin{align}
max quad & begin{pmatrix}0 \ 0 \ vdots \ 0 \ 1end{pmatrix}^T x' \
text{s.t.} quad & begin{pmatrix} A^T & -e \ e^T & 0 \ -e^T & 0end{pmatrix}x' leq begin{pmatrix} 0 \ 0 \ vdots \ 0 \ 1 \ -1end{pmatrix} \
& x' geq 0.
end{align}$$
Deriving the dual is now simple.
answered Nov 20 at 0:36
LinAlg
7,9241520
add a comment |
up vote
1
down vote
accepted
You could define $x'=begin{pmatrix}x \ zend{pmatrix}$, then the problem is:
$$begin{align}
max quad & begin{pmatrix}0 \ 0 \ vdots \ 0 \ 1end{pmatrix}^T x' \
text{s.t.} quad & begin{pmatrix} A^T & -e \ e^T & 0 \ -e^T & 0end{pmatrix}x' leq begin{pmatrix} 0 \ 0 \ vdots \ 0 \ 1 \ -1end{pmatrix} \
& x' geq 0.
end{align}$$
Deriving the dual is now simple.
answered Nov 20 at 0:36
LinAlg
7,9241520
add a comment |
up vote
1
down vote
accepted
up vote
1
down vote
accepted
You could define $x'=begin{pmatrix}x \ zend{pmatrix}$, then the problem is:
$$begin{align}
max quad & begin{pmatrix}0 \ 0 \ vdots \ 0 \ 1end{pmatrix}^T x' \
text{s.t.} quad & begin{pmatrix} A^T & -e \ e^T & 0 \ -e^T & 0end{pmatrix}x' leq begin{pmatrix} 0 \ 0 \ vdots \ 0 \ 1 \ -1end{pmatrix} \
& x' geq 0.
end{align}$$
Deriving the dual is now simple.
answered Nov 20 at 0:36
LinAlg
7,9241520
You could define $x'=begin{pmatrix}x \ zend{pmatrix}$, then the problem is:
$$begin{align}
max quad & begin{pmatrix}0 \ 0 \ vdots \ 0 \ 1end{pmatrix}^T x' \
text{s.t.} quad & begin{pmatrix} A^T & -e \ e^T & 0 \ -e^T & 0end{pmatrix}x' leq begin{pmatrix} 0 \ 0 \ vdots \ 0 \ 1 \ -1end{pmatrix} \
& x' geq 0.
end{align}$$
Deriving the dual is now simple.
answered Nov 20 at 0:36
LinAlg
7,9241520
answered Nov 20 at 0:36
LinAlg
7,9241520
answered Nov 20 at 0:36
LinAlg
7,9241520
answered Nov 20 at 0:36
LinAlg
7,9241520
7,9241520
add a comment |
add a comment |
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