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Show that the dual of $min c^Tx+d^Tx^'$ and $max a^Tx+b^Tx^'$ are equivalent

1

$begingroup$

Show that $(1)$ can be written in the form $(2)$

$(1)$
$min c^Tx+d^Tx^'$

$operatornames.t.$

$Ax+Bx^'geq a$

$Cx+Dx^'= b$
where $x, x^' geq 0$ and

and

$(2)$ $max a^Ty+b^Ty^'$

$operatornames.t.$

$A^Ty+C^Ty^'leq c$

$By+Dy^'= d$
where $y,y^' geq 0$

My idea:

The restrictions $(1)$ can be written in the form:

$mathcalP(beginpmatrix
A & -B \
C & D \
-C &-D
endpmatrix,beginpmatrix -a \ b \ -b endpmatrix)$

Then the dual LP to $(1)$ can be written as:

$max beginpmatrix a & -b & b endpmatrix^T(y, -y)$

$operatornames.t.$

$beginpmatrix
A & -B \
C & D \
-C &-D
endpmatrix^T(y,y^')=-beginpmatrix c \ d endpmatrix$

But how do I get to writing this in the form of $(2)$

Any help is greatly appreciated.

optimization linear-programming duality-theorems

share|cite|improve this question

edited Mar 28 at 11:33

SABOY

asked Mar 28 at 11:10

SABOYSABOY

592311

$endgroup$

This question has an open bounty worth +50
reputation from SABOY ending ending at 2019-04-06 11:18:38Z">in 4 days.

Looking for an answer drawing from credible and/or official sources.

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Do you want to cast (1) in the form (2) or do you want to cast the dual of (1) in the form (2)?
  $endgroup$
  – gerw
  Mar 28 at 11:32
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  dual of $(1)$ in the form $(2)$
  $endgroup$
  – SABOY
  Mar 28 at 11:33
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  you should check the dimensions, the objective of the dual has 3 coefficients in the objective but just two variables.
  $endgroup$
  – LinAlg
  2 days ago
  

  
  
  
  

add a comment | 

1

$begingroup$

Show that $(1)$ can be written in the form $(2)$

$(1)$
$min c^Tx+d^Tx^'$

$operatornames.t.$

$Ax+Bx^'geq a$

$Cx+Dx^'= b$
where $x, x^' geq 0$ and

and

$(2)$ $max a^Ty+b^Ty^'$

$operatornames.t.$

$A^Ty+C^Ty^'leq c$

$By+Dy^'= d$
where $y,y^' geq 0$

My idea:

The restrictions $(1)$ can be written in the form:

$mathcalP(beginpmatrix
A & -B \
C & D \
-C &-D
endpmatrix,beginpmatrix -a \ b \ -b endpmatrix)$

Then the dual LP to $(1)$ can be written as:

$max beginpmatrix a & -b & b endpmatrix^T(y, -y)$

$operatornames.t.$

$beginpmatrix
A & -B \
C & D \
-C &-D
endpmatrix^T(y,y^')=-beginpmatrix c \ d endpmatrix$

But how do I get to writing this in the form of $(2)$

Any help is greatly appreciated.

optimization linear-programming duality-theorems

share|cite|improve this question

edited Mar 28 at 11:33

SABOY

asked Mar 28 at 11:10

SABOYSABOY

592311

$endgroup$

This question has an open bounty worth +50
reputation from SABOY ending ending at 2019-04-06 11:18:38Z">in 4 days.

Looking for an answer drawing from credible and/or official sources.

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Do you want to cast (1) in the form (2) or do you want to cast the dual of (1) in the form (2)?
  $endgroup$
  – gerw
  Mar 28 at 11:32
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  dual of $(1)$ in the form $(2)$
  $endgroup$
  – SABOY
  Mar 28 at 11:33
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  you should check the dimensions, the objective of the dual has 3 coefficients in the objective but just two variables.
  $endgroup$
  – LinAlg
  2 days ago
  

  
  
  
  

add a comment | 

$begingroup$

Show that $(1)$ can be written in the form $(2)$

$(1)$
$min c^Tx+d^Tx^'$

$operatornames.t.$

$Ax+Bx^'geq a$

$Cx+Dx^'= b$
where $x, x^' geq 0$ and

and

$(2)$ $max a^Ty+b^Ty^'$

$operatornames.t.$

$A^Ty+C^Ty^'leq c$

$By+Dy^'= d$
where $y,y^' geq 0$

My idea:

The restrictions $(1)$ can be written in the form:

$mathcalP(beginpmatrix
A & -B \
C & D \
-C &-D
endpmatrix,beginpmatrix -a \ b \ -b endpmatrix)$

Then the dual LP to $(1)$ can be written as:

$max beginpmatrix a & -b & b endpmatrix^T(y, -y)$

$operatornames.t.$

$beginpmatrix
A & -B \
C & D \
-C &-D
endpmatrix^T(y,y^')=-beginpmatrix c \ d endpmatrix$

But how do I get to writing this in the form of $(2)$

Any help is greatly appreciated.

optimization linear-programming duality-theorems

share|cite|improve this question

edited Mar 28 at 11:33

SABOY

asked Mar 28 at 11:10

SABOYSABOY

592311

$endgroup$

share|cite|improve this question

edited Mar 28 at 11:33

SABOY

asked Mar 28 at 11:10

SABOYSABOY

592311

share|cite|improve this question

edited Mar 28 at 11:33

SABOY

asked Mar 28 at 11:10

SABOYSABOY

592311

asked Mar 28 at 11:10

SABOYSABOY

592311

asked Mar 28 at 11:10

SABOYSABOY

592311