Slack Variable Tutorial
Main.SlackVariables History
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{$\begin{align}\min \quad & cost_{total}\\\mathrm{subject\;to} \quad & supply<b\\& cost_{total}=(supply-2)^2\end{align}$}
'''Gekko (Python) Solution'''
(:source lang=python:)
# Solve slack variable problem
# Minimize total_cost
# Subject to supply < b
from gekko import GEKKO
b = 5
m = GEKKO(remote=False)
supply = m.Var()
total_cost = m.Var()
m.Equation(supply<b)
m.Equation(total_cost==(supply-2)**2)
m.Minimize(total_cost)
m.solve()
print(supply.value[0])
print(total_cost.value[0])
(:sourceend:)
'''APMonitor Solution'''
Attach:table50.jpg [[https://apmonitor.com/online/view_pass.php?f=slack.apm | Click to Solve Slack Variable Optimization Problem Online]]
'''Gekko (Python) Solution'''
(:source lang=python:)
# Solve slack variable problem
# Minimize total_cost
# Subject to supply < b
from gekko import GEKKO
b = 5
m = GEKKO(remote=False)
supply = m.Var()
total_cost = m.Var()
m.Equation(supply<b)
m.Equation(total_cost==(supply-2)**2)
m.Minimize(total_cost)
m.solve()
print(supply.value[0])
print(total_cost.value[0])
(:sourceend:)
'''APMonitor Solution'''
Attach:table50.jpg [[https://apmonitor.com/online/view_pass.php?f=slack.apm | Click to Solve Slack Variable Optimization Problem Online]]
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!!! Inequality Constraints in APM
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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<iframe width="560" height="315" src="https://www.youtube.com/embed/NPUNixLGlh0" frameborder="0" allowfullscreen></iframe>
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In APMonitor Modeling Language, inequality constraints are automatically translated into equality constraints with slack variables. Slack
!!! 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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----
!!! Inequality Constraints
In [[https://gekko.readthedocs.io/en/latest/|Gekko Optimization Suite]] and the [[https://apmonitor.com/wiki/|APMonitor Modeling Language]], inequality constraints are automatically translated into equality constraints with slack variables. In APMonitor, 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 are automatically converted to equality constraints with a slack variable.
!!! Inequality Constraints
In [[https://gekko.readthedocs.io/en/latest/|Gekko Optimization Suite]] and the [[https://apmonitor.com/wiki/|APMonitor Modeling Language]], inequality constraints are automatically translated into equality constraints with slack variables. In APMonitor, 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 are 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.
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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.
(:html:)
<iframe width="560" height="315" src="https://www.youtube.com/embed/NPUNixLGlh0" frameborder="0" allowfullscreen></iframe>
(:htmlend:)
!!! 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.
(:html:)
<iframe width="560" height="315" src="https://www.youtube.com/embed/NPUNixLGlh0" frameborder="0" allowfullscreen></iframe>
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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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(:tableend:)
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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(:html:)
<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.