Slack Variable Tutorial
Main.SlackVariables History
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In APMonitor Modeling Language, inequality constraints are automatically translated into equality constraints with slack variables. Slack variables can also be defined by starting a variable name with ''slk''. When the model is parsed at run-time, any variable beginning with ''slk'' is automatically assigned a lower value of zero. Alternatively, inequality constraints will be automatically converted to equality constraints with a slack variable.
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In APMonitor Modeling Language, inequality constraints are automatically translated into equality constraints with slack variables. Slack variables can also be defined by starting a variable name with ''slk''. When the model is parsed at run-time, any variable beginning with ''slk'' is automatically assigned a lower value of zero. Alternatively, inequality constraints will be automatically converted to equality constraints with a slack variable.
!!! Logical Conditions in Optimization
Slack variables are also useful for optimization problems with non-smooth or discontinuous functions. The following tutorial demonstrates the use of slack variables in reformulating the ABS() function, IF statements, and others.
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!!! Logical Conditions in Optimization
Slack variables are also useful for optimization problems with non-smooth or discontinuous functions. The following tutorial demonstrates the use of slack variables in reformulating the ABS() function, IF statements, and others.
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! Example model with slack variable
! Solve slack variable problem
! Minimize total_cost
! Subject to supply < b
Model
Parameters
b = 5
End Parameters
Variables
supply
total_cost
End Variables
Equations
supply < b
total_cost = (supply-2)^2
Minimize total_cost
End Equations
End Model
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Attach:slack_problem.png
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!! Example Problem
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----
!!! Example Problem
!!! Example Problem
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!! Inequality Constraints in APM
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----
!!! Inequality Constraints in APM
!!! Inequality Constraints in APM
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!! Example Problem
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!! Inequality Constraints in APM
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Attach:table50.jpg [[https://apmonitor.com/online/view_pass.php?f=slack.apm | Click to Solve a Slack Variable Optimization Problem]]
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In APMonitor Modeling Language, inequality constraints are automatically translated into equality constraints with slack variables. Slack variables can also be defined by starting a variable name with ''slk''. When the model is parsed at run-time, any variable beginning with ''slk'' is automatically assigned a lower value of zero. Alternatively, inequality constraints will be automatically converted to equality constraints with a slack variable.
In APMonitor Modeling Language, inequality constraints are automatically translated into equality constraints with slack variables. Slack variables can also be defined by starting a variable name with ''slk''. When the model is parsed at run-time, any variable beginning with ''slk'' is automatically assigned a lower value of zero. Alternatively, inequality constraints will be automatically converted to equality constraints with a slack variable.
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Slack variables are defined to transform an inequality expression into an equality expression with an added slack variable. The slack variable is defined by setting a lower bound of zero (>=0).
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Slack variables are defined to transform an inequality expression into an equality expression with an added slack variable. The slack variable is defined by setting a lower bound of zero (>0).
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slack >= 0
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slack > 0
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<iframe width="560" height="315" src="https://www.youtube.com/embed/jh6BK0BqqIs?rel=0" frameborder="0" allowfullscreen></iframe>
(:htmlend:)
In APMonitor, inequality constraints are automatically translated into equality constraints with slack variables. Slack variables can also be defined by starting a variable name with ''slk''. When the model is parsed at run-time, any variable beginning with ''slk'' is automatically assigned a lower value of zero. Alternatively, inequality constraints will be automatically converted to equality constraints with a slack variable.
<iframe width="560" height="315" src="https://www.youtube.com/embed/jh6BK0BqqIs?rel=0" frameborder="0" allowfullscreen></iframe>
(:htmlend:)
In APMonitor, inequality constraints are automatically translated into equality constraints with slack variables. Slack variables can also be defined by starting a variable name with ''slk''. When the model is parsed at run-time, any variable beginning with ''slk'' is automatically assigned a lower value of zero. Alternatively, inequality constraints will be automatically converted to equality constraints with a slack variable.
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Attach:table50.jpg [[https://apmonitor.com/online/view_pass.php?f=slack.apm | Solve Slack Variable Optimization Problems]]
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Attach:table50.jpg [[https://apmonitor.com/online/view_pass.php?f=slack.apm | Click to Solve a Slack Variable Optimization Problem]]
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(:title Slack Variable Tutorial:)
(:keywords slack variables, inequality, slacks, lower bound, transform:)
(:description Slack variables are used to transform an inequality expression into an equality expression:)
(:keywords slack variables, inequality, slacks, lower bound, transform:)
(:description Slack variables are used to transform an inequality expression into an equality expression:)
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Attach:table50.jpg [[https://apmonitor.com/online/view_pass.php?f=slack.apm | Solve Slack Variable Optimization Problems]]
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!!! Solve Slack Variable Problems Online
%blue%A%red%P%black%Monitor has an online interface for solving optimization problems. The optimization problem shown below as '''Example Problem''' can be solved through the web-interface at the following link:
Attach:table50.jpg [[https://apmonitor.com/online/view_pass.php?f=slack.apm | Slack Variable Solution Online]]
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x = b + slack
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In APMonitor, slack variables can be defined by starting a variable name with '''slk'''. When the model is parsed at run-time, any variable beginning with ''slk'' is automatically assigned a lower value of zero. Alternatively, inequality constraints will be automatically converted to equality constraints with a slack variable.
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In APMonitor, slack variables can be defined by starting a variable name with ''slk''. When the model is parsed at run-time, any variable beginning with ''slk'' is automatically assigned a lower value of zero. Alternatively, inequality constraints will be automatically converted to equality constraints with a slack variable.
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In APMonitor, slack variables can be defined by starting a variable name with ''''slk''''. When the model is parsed at run-time, any variable beginning with ''slk'' is automatically assigned a lower value of zero. Alternatively, inequality constraints will be automatically converted to equality constraints with a slack variable.
to:
In APMonitor, slack variables can be defined by starting a variable name with '''slk'''. When the model is parsed at run-time, any variable beginning with ''slk'' is automatically assigned a lower value of zero. Alternatively, inequality constraints will be automatically converted to equality constraints with a slack variable.
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Equation inequalities are automatically tranformed to equality expressions for solvers that do not support them
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(:cellnr:)
Inequality Constraint Form
(:cell:)
x > b
(:cellnr:)
Equality Constraint Form with Slack Variable
(:cell:)
x = b + slack
slack >= 0
(:tableend:)
In APMonitor, slack variables can be defined by starting a variable name with ''''slk''''. When the model is parsed at run-time, any variable beginning with ''slk'' is automatically assigned a lower value of zero. Alternatively, inequality constraints will be automatically converted to equality constraints with a slack variable.
(:cellnr:)
Inequality Constraint Form
(:cell:)
x > b
(:cellnr:)
Equality Constraint Form with Slack Variable
(:cell:)
x = b + slack
slack >= 0
(:tableend:)
In APMonitor, slack variables can be defined by starting a variable name with ''''slk''''. When the model is parsed at run-time, any variable beginning with ''slk'' is automatically assigned a lower value of zero. Alternatively, inequality constraints will be automatically converted to equality constraints with a slack variable.
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! Example model with slack variable declaration
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! Example model with slack variable
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Parameters
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! Solve slack variable problem
! Minimize total_cost
! Subject to supply < b
Model
Parameters
b = 5
End Parameters
Variables
supply
total_cost
End Variables
Equations
supply < b
total_cost = (supply-2)^2
Minimize total_cost
End Equations
! Minimize total_cost
! Subject to supply < b
Model
Parameters
b = 5
End Parameters
Variables
supply
total_cost
End Variables
Equations
supply < b
total_cost = (supply-2)^2
Minimize total_cost
End Equations
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p1 = 5
slk
v1 = 4
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Attach:slack_results.png
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Slack variables are defined to transform a inequality expression into an equality expression with an added slack variable. The slack variable is defined by setting a lower bound of zero (>=0).
to:
Slack variables are defined to transform an inequality expression into an equality expression with an added slack variable. The slack variable is defined by setting a lower bound of zero (>=0).
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%blue%A%red%P%black%Monitor has an online interface for solving optimization problems. The optimization problem shown below can be solved through the web-interface at the following link:
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%blue%A%red%P%black%Monitor has an online interface for solving optimization problems. The optimization problem shown below as '''Example Problem''' can be solved through the web-interface at the following link:
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!!! Example
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!!! Example Problem
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Attach: table50.jpg [[https://apmonitor.com/online/view_pass.php?f=slack.apm | Slack Variable Solution Online]]
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Attach:table50.jpg [[https://apmonitor.com/online/view_pass.php?f=slack.apm | Slack Variable Solution Online]]
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[[https://apmonitor.com/online/view_pass.php?f=slack.apm | Slack Variable Solution Online]]
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Attach: table50.jpg [[https://apmonitor.com/online/view_pass.php?f=slack.apm | Slack Variable Solution Online]]
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%blue%A%red%P%black%Monitor has an online interface for solving optimization problems. The optimization problem shown below can be solved the web-interface at the following link:
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%blue%A%red%P%black%Monitor has an online interface for solving optimization problems. The optimization problem shown below can be solved through the web-interface at the following link:
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!!! Solve Slack Variable Problems Online
%blue%A%red%P%black%Monitor has an online interface for solving optimization problems. The optimization problem shown below can be solved the web-interface at the following link:
[[https://apmonitor.com/online/view_pass.php?f=slack.apm | Slack Variable Solution Online]]
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Slack variables are defined to give ''slack'' for infeasibilities. Generally, a slack variable is used to turn off an equation in a region where it does not apply. The slack variable is defined by setting a lower bound of zero (>=0) and including the variable in the objective function.
Slack variables are defined by starting a variable name with''slk''''. Thus, valid slack variable names would include names such as ''slk_1'', ''slkr'', ''slkvar[1]''. When the model is parsed at run-time, any variable beginning with ''slk'' is automatically assigned a lower value of zero and included in the objective function.
Slack variables are defined by starting a variable name with
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Slack variables are defined to transform a inequality expression into an equality expression with an added slack variable. The slack variable is defined by setting a lower bound of zero (>=0).
Slack variables are defined by starting a variable name with ''''slk''''. Thus, valid slack variable names would include names such as ''slk_1'', ''slkr'', ''slkvar[1]''. When the model is parsed at run-time, any variable beginning with ''slk'' is automatically assigned a lower value of zero.
Equation inequalities are automatically tranformed to equality expressions for solvers that do not support them.
Slack variables are defined by starting a variable name with ''''slk''''. Thus, valid slack variable names would include names such as ''slk_1'', ''slkr'', ''slkvar[1]''. When the model is parsed at run-time, any variable beginning with ''slk'' is automatically assigned a lower value of zero.
Equation inequalities are automatically tranformed to equality expressions for solvers that do not support them.
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!! Slack Variables
Slack variables are defined to give ''slack'' for infeasibilities. Generally, a slack variable is used to turn off an equation in a region where it does not apply. The slack variable is defined by setting a lower bound of zero (>=0) and including the variable in the objective function.
Slack variables are defined by starting a variable name with ''''slk''''. Thus, valid slack variable names would include names such as ''slk_1'', ''slkr'', ''slkvar[1]''. When the model is parsed at run-time, any variable beginning with ''slk'' is automatically assigned a lower value of zero and included in the objective function.
!!! Example
(:table border=1 width=50% align=left bgcolor=#EEEEEE cellspacing=0:)
(:cellnr:)
! Example model with slack variable declaration
Model example
Parameters
p1 = 5
End Parameters
Variables
slk_variable
v1 < 4
End Variables
Equations
v1 = p1 - slk_variable
End Equations
End Model
(:cellnr:)
Solution
p1 = 5
slk_variable = 1
v1 = 4
(:tableend:)
Slack variables are defined to give ''slack'' for infeasibilities. Generally, a slack variable is used to turn off an equation in a region where it does not apply. The slack variable is defined by setting a lower bound of zero (>=0) and including the variable in the objective function.
Slack variables are defined by starting a variable name with ''''slk''''. Thus, valid slack variable names would include names such as ''slk_1'', ''slkr'', ''slkvar[1]''. When the model is parsed at run-time, any variable beginning with ''slk'' is automatically assigned a lower value of zero and included in the objective function.
!!! Example
(:table border=1 width=50% align=left bgcolor=#EEEEEE cellspacing=0:)
(:cellnr:)
! Example model with slack variable declaration
Model example
Parameters
p1 = 5
End Parameters
Variables
slk_variable
v1 < 4
End Variables
Equations
v1 = p1 - slk_variable
End Equations
End Model
(:cellnr:)
Solution
p1 = 5
slk_variable = 1
v1 = 4
(:tableend:)