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Slack Variable Tutorial

Slack Variables

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).

Inequality Constraint Form

x > b

Equality Constraint Form with Slack Variable

x = b + slack
slack >= 0






Click to Solve a Slack Variable Optimization Problem

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.

Example Problem

 ! 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