Design Optimization

or with the slack variable

where ''s'' is the slack variable.

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'').

{$2x + y + s = 4$}

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).

slack > 0

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).

Slack variables are additional variables used in optimization that convert an inequality constraint into an equality constraint, allowing the problem to be solved with standard methods, such as [[Main/InteriorPointMethod|Interior Point Methods]]. Slack variables are typically denoted by the letter ''s'' and the value is always positive. Slack variables convert a constraint of the form "greater than or equal to" into an equation. For example, if the constraint is ''2x + y ≥ 4'', then the constraint can be rewritten as ''2x + y + s = 4'', where ''s'' is the slack variable.

'''Gekko Solution'''

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. 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.

!!! 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.

{$\begin{align}\min \quad & cost_{total}\\\mathrm{subject to} \quad & supply<b\\& cost_{total}=(supply-2)^2\end{align}$}

{$\begin{align}\min & cost_{total}\\subject to & supply<b\\& cost_{total}=(supply-2)^2\end{align}$}

Attach:table50.jpg [[https://apmonitor.com/online/view_pass.php?f=slack.apm | Click to Solve a Slack Variable Optimization Problem]]