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Standard cumulative distribution function with optimization model variable

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3

$begingroup$

We all know that expressions in mathematical optimization models can't contain "black boxes" around a decision variable since everything has to be written using mathematical expressions. For example, "yes/no" decisions can't be written as "if this then that" expressions in a model, but can be written using big-$M$ constraints with a binary variable.

What is the recommended way of writing a model that must compute the value of a cumulative distribution function from a decision variable?

For example, let's suppose we have a model with the following constraint:

$$Phi_X(x)le b$$

where $Phi_X(x)$ is the cumulative distribution function of the standard Normal random variable $Xsim N(mu = 0,sigma =1)$, $x$ is a decision variable, and $b$ is a parameter of the model.

Is there any way of converting this kind of constraint into a valid mathematical formulation? What is the approach suggested by experts? Does that kind of constraint absolutely require the use of special tools that can't be found in regular optimization solvers?

optimization modeling probability-distributions

share|improve this question

edited 1 hour ago

SecretAgentMan

1,11222 silver badges2323 bronze badges

asked 8 hours ago

V. BrunelleV. Brunelle

16177 bronze badges

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  What do you men by "characteristic function of the standard deviation", care to explain?
  $endgroup$
  – kjetil b halvorsen
  5 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Is it me or English lacks this definition? "Loi normale centrée réduite" (in French) is what I mean. The "default" standard law, when the average is 0 and the standard error is 1.
  $endgroup$
  – V. Brunelle
  4 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I do not speak french, so let us wait for some french speaker to show up ...
  $endgroup$
  – kjetil b halvorsen
  4 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Is $Phi(x)$ the CDF? As in $Phi_X(x) = P(Xle x)$ for the standard Normal distribution for $Xsim N(mu =0, sigma = 1)$, $text{E}[X]=mu$ and $sqrt{text{Var}(X)}=sigma$?
  $endgroup$
  – SecretAgentMan
  2 hours ago
  
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @SecretAgentMan Yes it is. µ=0 and sigma=1.
  $endgroup$
  – V. Brunelle
  2 hours ago
  

  
  
  
  

add a comment
 | 

3

$begingroup$

We all know that expressions in mathematical optimization models can't contain "black boxes" around a decision variable since everything has to be written using mathematical expressions. For example, "yes/no" decisions can't be written as "if this then that" expressions in a model, but can be written using big-$M$ constraints with a binary variable.

What is the recommended way of writing a model that must compute the value of a cumulative distribution function from a decision variable?

For example, let's suppose we have a model with the following constraint:

$$Phi_X(x)le b$$

where $Phi_X(x)$ is the cumulative distribution function of the standard Normal random variable $Xsim N(mu = 0,sigma =1)$, $x$ is a decision variable, and $b$ is a parameter of the model.

Is there any way of converting this kind of constraint into a valid mathematical formulation? What is the approach suggested by experts? Does that kind of constraint absolutely require the use of special tools that can't be found in regular optimization solvers?

optimization modeling probability-distributions

share|improve this question

edited 1 hour ago

SecretAgentMan

1,11222 silver badges2323 bronze badges

asked 8 hours ago

V. BrunelleV. Brunelle

16177 bronze badges

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  What do you men by "characteristic function of the standard deviation", care to explain?
  $endgroup$
  – kjetil b halvorsen
  5 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Is it me or English lacks this definition? "Loi normale centrée réduite" (in French) is what I mean. The "default" standard law, when the average is 0 and the standard error is 1.
  $endgroup$
  – V. Brunelle
  4 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I do not speak french, so let us wait for some french speaker to show up ...
  $endgroup$
  – kjetil b halvorsen
  4 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Is $Phi(x)$ the CDF? As in $Phi_X(x) = P(Xle x)$ for the standard Normal distribution for $Xsim N(mu =0, sigma = 1)$, $text{E}[X]=mu$ and $sqrt{text{Var}(X)}=sigma$?
  $endgroup$
  – SecretAgentMan
  2 hours ago
  
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @SecretAgentMan Yes it is. µ=0 and sigma=1.
  $endgroup$
  – V. Brunelle
  2 hours ago
  

  
  
  
  

add a comment
 | 

$begingroup$

We all know that expressions in mathematical optimization models can't contain "black boxes" around a decision variable since everything has to be written using mathematical expressions. For example, "yes/no" decisions can't be written as "if this then that" expressions in a model, but can be written using big-$M$ constraints with a binary variable.

What is the recommended way of writing a model that must compute the value of a cumulative distribution function from a decision variable?

For example, let's suppose we have a model with the following constraint:

$$Phi_X(x)le b$$

where $Phi_X(x)$ is the cumulative distribution function of the standard Normal random variable $Xsim N(mu = 0,sigma =1)$, $x$ is a decision variable, and $b$ is a parameter of the model.

Is there any way of converting this kind of constraint into a valid mathematical formulation? What is the approach suggested by experts? Does that kind of constraint absolutely require the use of special tools that can't be found in regular optimization solvers?

optimization modeling probability-distributions

share|improve this question

edited 1 hour ago

SecretAgentMan

1,11222 silver badges2323 bronze badges

asked 8 hours ago

V. BrunelleV. Brunelle

16177 bronze badges

$endgroup$

share|improve this question

edited 1 hour ago

SecretAgentMan

1,11222 silver badges2323 bronze badges

asked 8 hours ago

V. BrunelleV. Brunelle

16177 bronze badges

share|improve this question

edited 1 hour ago

SecretAgentMan

1,11222 silver badges2323 bronze badges

asked 8 hours ago

V. BrunelleV. Brunelle

16177 bronze badges

edited 1 hour ago

SecretAgentMan

1,11222 silver badges2323 bronze badges

edited 1 hour ago

SecretAgentMan

1,11222 silver badges2323 bronze badges

asked 8 hours ago

V. BrunelleV. Brunelle

16177 bronze badges

asked 8 hours ago

V. BrunelleV. Brunelle

16177 bronze badges

1 Answer
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7

$begingroup$

For strictly increasing CDFs, you can invert:
$$x le Phi^{-1}(b)$$

share|improve this answer

answered 8 hours ago

Rob PrattRob Pratt

1,46711 silver badge1212 bronze badges

$endgroup$

add a comment
 | 

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7

$begingroup$

For strictly increasing CDFs, you can invert:
$$x le Phi^{-1}(b)$$

share|improve this answer

answered 8 hours ago

Rob PrattRob Pratt

1,46711 silver badge1212 bronze badges

$endgroup$

add a comment
 | 

7

$begingroup$

For strictly increasing CDFs, you can invert:
$$x le Phi^{-1}(b)$$

share|improve this answer

answered 8 hours ago

Rob PrattRob Pratt

1,46711 silver badge1212 bronze badges

$endgroup$

add a comment
 | 

$begingroup$

For strictly increasing CDFs, you can invert:
$$x le Phi^{-1}(b)$$

share|improve this answer

answered 8 hours ago

Rob PrattRob Pratt

1,46711 silver badge1212 bronze badges

$endgroup$

share|improve this answer

answered 8 hours ago

Rob PrattRob Pratt

1,46711 silver badge1212 bronze badges

answered 8 hours ago

Rob PrattRob Pratt

1,46711 silver badge1212 bronze badges

answered 8 hours ago

Rob PrattRob Pratt

1,46711 silver badge1212 bronze badges