Slack variable

In an optimization problem, a slack variable is a variable that is added to an inequality constraint to transform it to an equality. Introducing a slack variable replaces an inequality constraint with an equality constraint and a nonnegativity constraint.

In linear programming, this is required to turn an inequality into an equality where a linear combination of variables is less than or equal to a given constant in the former. As with the other variables in the augmented constraints, the slack variable cannot take on negative values, as the Simplex algorithm requires them to be positive or zero.

By introducing the slack variable ${\displaystyle \mathbf {y} \geq \mathbf {0} }$, the inequality ${\displaystyle \mathbf {A} \mathbf {x} \leq \mathbf {b} }$ can be converted to the equation ${\displaystyle \mathbf {A} \mathbf {x} +\mathbf {y} =\mathbf {b} }$.

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