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Why is one optimal value greater than or equal to the other one here?

1

$begingroup$

Let the first program be $$min frac{c^Tx + d}{e^Tx+f} text{ subject to }{Gx le h, Ax = b}$$

It can be transformed to equivalent linear program:

$$min (c^Ty + dz) text{ subject to } {Gy - hz ge 0, Ay - bz = 0, e^Ty + fz = 1, z ge 0}$$

If $x$ is feasible in the first one, then $y = frac{x}{e^Tx + f}, z = frac{1}{e^Tx + f}$ is feasible in the second one. It follows that the optimal value of the original program is greater than or equal to the optimal value of the transformed program.

Can someone explain why the optimal value of the original program is greater than or equal to the optimal value of the transformed program?

proof-explanation convex-optimization

share|cite|improve this question

edited Mar 24 at 20:11

Oliver G

asked Mar 24 at 19:57

Oliver GOliver G

1,2761634

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  This just reflects the fact that you're only proving one direction of the if-and-only-if equivalence. You're proving that every feasible point in the first problem leads to a feasible point in the second—but you're not saying anything about optimality. So given an optimal points for the first problem, you've only proved that leads to a feasible, but possibly suboptimal, point in the second. Without further proof, it's possible that the optimal point for the second problem cannot be mapped back to the first.
  $endgroup$
  – Michael Grant
  Mar 25 at 3:25
  
  
  

  
  
  
  

add a comment | 

1

$begingroup$

Let the first program be $$min frac{c^Tx + d}{e^Tx+f} text{ subject to }{Gx le h, Ax = b}$$

It can be transformed to equivalent linear program:

$$min (c^Ty + dz) text{ subject to } {Gy - hz ge 0, Ay - bz = 0, e^Ty + fz = 1, z ge 0}$$

If $x$ is feasible in the first one, then $y = frac{x}{e^Tx + f}, z = frac{1}{e^Tx + f}$ is feasible in the second one. It follows that the optimal value of the original program is greater than or equal to the optimal value of the transformed program.

Can someone explain why the optimal value of the original program is greater than or equal to the optimal value of the transformed program?

proof-explanation convex-optimization

share|cite|improve this question

edited Mar 24 at 20:11

Oliver G

asked Mar 24 at 19:57

Oliver GOliver G

1,2761634

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  This just reflects the fact that you're only proving one direction of the if-and-only-if equivalence. You're proving that every feasible point in the first problem leads to a feasible point in the second—but you're not saying anything about optimality. So given an optimal points for the first problem, you've only proved that leads to a feasible, but possibly suboptimal, point in the second. Without further proof, it's possible that the optimal point for the second problem cannot be mapped back to the first.
  $endgroup$
  – Michael Grant
  Mar 25 at 3:25
  
  
  

  
  
  
  

add a comment | 

$begingroup$

Let the first program be $$min frac{c^Tx + d}{e^Tx+f} text{ subject to }{Gx le h, Ax = b}$$

It can be transformed to equivalent linear program:

$$min (c^Ty + dz) text{ subject to } {Gy - hz ge 0, Ay - bz = 0, e^Ty + fz = 1, z ge 0}$$

If $x$ is feasible in the first one, then $y = frac{x}{e^Tx + f}, z = frac{1}{e^Tx + f}$ is feasible in the second one. It follows that the optimal value of the original program is greater than or equal to the optimal value of the transformed program.

Can someone explain why the optimal value of the original program is greater than or equal to the optimal value of the transformed program?

proof-explanation convex-optimization

share|cite|improve this question

edited Mar 24 at 20:11

Oliver G

asked Mar 24 at 19:57

Oliver GOliver G

1,2761634

$endgroup$

share|cite|improve this question

edited Mar 24 at 20:11

Oliver G

asked Mar 24 at 19:57

Oliver GOliver G

1,2761634

share|cite|improve this question

edited Mar 24 at 20:11

Oliver G

asked Mar 24 at 19:57

Oliver GOliver G

1,2761634

asked Mar 24 at 19:57

Oliver GOliver G

1,2761634

asked Mar 24 at 19:57

Oliver GOliver G

1,2761634