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In linear programming, what does the primal problem imply about the sign of the dual variables?

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$begingroup$

In Bertsimas' Introduction to Linear Optimization, we define the primal problem in standard form as minimizing $c'x$ subject to $Ax=b$ and $xgeq0$, and the dual problem as maximizing $p'b$ subject to $A^* p geq c$.

He makes the comment that this definition of the dual problem requires $p_i$ to have the same sign as that of $a'_i x - b_i$. I've gone back through the derivation of the dual problem and its constraints a few time, and am having trouble understanding why this is.

Any thoughts?

Thanks!

linear-programming

share|cite|improve this question

edited Mar 11 at 19:56

Brian

601114

asked Mar 11 at 19:50

user49404user49404

579

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  A good explanation about transforming a primal problem into a dual problem can be found here by Mike Spivey
  $endgroup$
  – callculus
  Mar 11 at 20:23
  
  

  
  
  
  

add a comment | 

0

$begingroup$

In Bertsimas' Introduction to Linear Optimization, we define the primal problem in standard form as minimizing $c'x$ subject to $Ax=b$ and $xgeq0$, and the dual problem as maximizing $p'b$ subject to $A^* p geq c$.

He makes the comment that this definition of the dual problem requires $p_i$ to have the same sign as that of $a'_i x - b_i$. I've gone back through the derivation of the dual problem and its constraints a few time, and am having trouble understanding why this is.

Any thoughts?

Thanks!

linear-programming

share|cite|improve this question

edited Mar 11 at 19:56

Brian

601114

asked Mar 11 at 19:50

user49404user49404

579

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  A good explanation about transforming a primal problem into a dual problem can be found here by Mike Spivey
  $endgroup$
  – callculus
  Mar 11 at 20:23
  
  

  
  
  
  

add a comment | 

$begingroup$

In Bertsimas' Introduction to Linear Optimization, we define the primal problem in standard form as minimizing $c'x$ subject to $Ax=b$ and $xgeq0$, and the dual problem as maximizing $p'b$ subject to $A^* p geq c$.

He makes the comment that this definition of the dual problem requires $p_i$ to have the same sign as that of $a'_i x - b_i$. I've gone back through the derivation of the dual problem and its constraints a few time, and am having trouble understanding why this is.

Any thoughts?

Thanks!

linear-programming

share|cite|improve this question

edited Mar 11 at 19:56

Brian

601114

asked Mar 11 at 19:50

user49404user49404

579

$endgroup$

share|cite|improve this question

edited Mar 11 at 19:56

Brian

601114

asked Mar 11 at 19:50

user49404user49404

579

share|cite|improve this question

edited Mar 11 at 19:56

Brian

601114

asked Mar 11 at 19:50

user49404user49404

579

edited Mar 11 at 19:56

Brian

601114

edited Mar 11 at 19:56

Brian

601114

asked Mar 11 at 19:50

user49404user49404

579

asked Mar 11 at 19:50

user49404user49404

579