In mathematics, Slater's condition (or Slater condition) is a sufficient condition for strong duality to hold for a convex optimization problem, named after Morton L.
[1] Informally, Slater's condition states that the feasible region must have an interior point (see technical details below).
Slater's condition is a specific example of a constraint qualification.
[2] 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.
be real-valued functions on some subset
We say that the functions satisfy the Slater condition if there exists some
in the relative interior of
We say that the functions satisfy the relaxed Slater condition if:[3] Consider the optimization problem where
are convex functions.
This is an instance of convex programming.
Slater's condition for convex programming states that there exists an
that is strictly feasible, that is, all m constraints are satisfied, and the nonlinear constraints are satisfied with strict inequalities.
If a convex program satisfies Slater's condition (or relaxed condition), and it is bounded from below, then strong duality holds.
Mathematically, this states that strong duality holds if there exists an
(where relint denotes the relative interior of the convex set
Then Slater's condition says that if there exists an
such that then strong duality holds.