Slater's condition
In mathematics, Slater's condition is a sufficient condition for strong duality to hold for a convex optimization problem, named after Morton L. Slater. Informally, Slater's condition states that the feasible region must have an interior point.
Slater's condition is a specific example of a constraint qualification. In particular, if Slater's condition holds for the primal problem, then the duality gap is 0, and if the dual value is finite then it is attained.Formulation
Consider the optimization problem
where are convex functions. This is an instance of convex programming.
In words, Slater's condition for convex programming states that strong duality holds if there exists an such that is strictly feasible.
Mathematically, Slater's condition states that strong duality holds if there exists an such thatGiven the problem
where is convex and is -convex for each. Then Slater's condition says that if there exists an such that
then strong duality holds.
