Slack variable
In an
Slack variables are used in particular in linear programming. 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.[2]
- If a slack variable associated with a constraint is zero at a particular bindingthere, as the constraint restricts the possible changes from that point.
- If a slack variable is positive at a particular candidate solution, the constraint is non-bindingthere, as the constraint does not restrict the possible changes from that point.
- If a slack variable is negative at some point, the point is infeasible (not allowed), as it does not satisfy the constraint.
Slack variables are also used in the Big M method.
Example
By introducing the slack variable , the inequality can be converted to the equation .
Embedding in orthant
Slack variables give an embedding of a polytope into the standard f-orthant, where 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
Dually, generalized barycentric coordinates express a polytope with vertices (dual to facets), regardless of dimension, as the image of the standard -simplex, which has vertices – the map is onto: 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.
References
- ISBN 978-0-521-83378-3. Retrieved October 15, 2011.
- ISBN 3-540-30697-8.: 42
External links
- Slack Variable Tutorial - Solve slack variable problems online