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.^[1]

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.

If a slack variable associated with a constraint is zero in a given state, the constraint is binding, as the constraint restricts the possible changes of the point.
If a slack variable is positive in a given state, the constraint is non-binding, as the constraint does not restrict the possible changes of the point.
If a slack variable is negative in a given state, the point is infeasible, and not allowed, as it does not satisfy the constraint.

Example

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

Embedding in orthant

Further information: Orthant and Generalized barycentric coordinates

Slack variables give an embedding of a polytope $P \hookrightarrow (\mathbf{R}_{\geq 0})^f$ into the standard f-orthant, where f is the number of constraints (facets of the polytope). This map is one-to-one (slack variables are uniquely determined) but not onto (not all combinations can be realized), and is expressed in terms of the constraints (linear functionals, covectors).

Slack variables are dual to generalized barycentric coordinates, and, dually to generalized barycentric coordinates (which are not unique but can all be realized), are uniquely determined, but cannot all be realized.

Dually, generalized barycentric coordinates express a polytope with n vertices (dual to facets), regardless of dimension, as the image of the standard $(n-1)$ -simplex, which has n vertices – the map is onto: $\Delta ^{{n-1}}\twoheadrightarrow P,$ and expresses points in terms of the vertices (points, vectors). The map is one-to-one if and only if the polytope is a simplex, in which case the map is an isomorphism; this corresponds to a point not having unique generalized barycentric coordinates.

Notes

↑ Boyd & Vandenberghe 2004, p. 131.

References

Boyd, Stephen P.; Vandenberghe, Lieven (2004). Convex Optimization (pdf). Cambridge University Press. ISBN 978-0-521-83378-3. Retrieved October 15, 2011.

External links

Slack Variable Tutorial - Solve slack variable problems online

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