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| {{More footnotes|date=October 2011}}
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| 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.{{sfn|Boyd|Vandenberghe|2004|p=131}}
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| 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.
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| * If a slack variable associated with a constraint is ''zero'' in a given state, the constraint is '''[[binding constraint|binding]]''', as the constraint restricts the possible changes of the point.
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| * If a slack variable is ''positive'' in a given state, the constraint is '''[[non-binding constraint|non-binding]]''', as the constraint does not restrict the possible changes of the point.
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| * If a slack variable is ''negative'' in a given state, the point is '''infeasible''', and not allowed, as it does not satisfy the constraint.
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| ==Example==
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| By introducing the slack variable <math>\mathbf{y} \ge 0</math>, the inequality
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| <math>\mathbf{A}\mathbf{x} \le \mathbf{b}</math> can be converted to the equation
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| <math>\mathbf{A}\mathbf{x} + \mathbf{y} = \mathbf{b}</math>.
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| == Embedding in orthant == | |
| {{further|Orthant|Generalized barycentric coordinates}}
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| Slack variables give an embedding of a polytope <math>P \hookrightarrow (\mathbf{R}_{\geq 0})^f</math> 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).
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| Slack variables are ''[[Dual linear program|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.
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| Dually, generalized barycentric coordinates express a polytope with ''n'' vertices (dual to facets), regardless of dimension, as the ''image'' of the standard <math>(n-1)</math>-simplex, which has ''n'' vertices – the map is onto: <math>\Delta^{n-1} \twoheadrightarrow P,</math> 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.
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| ==See also==
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| *[[Simplex algorithm]]
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| == Notes ==
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| {{reflist}}
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| ==References==
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| * {{cite book|title=Convex Optimization|first1=Stephen P.|last1=Boyd|first2=Lieven|last2=Vandenberghe|year=2004|publisher=Cambridge University Press|isbn=978-0-521-83378-3|url=http://www.stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf|format=pdf|accessdate=October 15, 2011|ref=harv}}
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| ==External links==
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| * [http://apmonitor.com/wiki/index.php/Main/SlackVariables Slack Variable Tutorial] - Solve slack variable problems online
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| [[Category:Mathematical optimization]]
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I am Oscar and I completely dig that name. Body developing is 1 of over the counter std test issues I love most. For many years he's been residing in North Dakota and his family members loves it. Supervising is my profession.