# 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.

• 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.

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... To each variable in the primal space corresponds an inequality to satisfy in the dual space, both indexed by output type ... satisfy in the primal space corresponds a variable in the dual space, both indexed by input type ... Since each inequality can be replaced by an equality and a slack variable, this means each primal variable corresponds to a dual slack variable, and each dual variable ...

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