# Slater's condition

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. Slater. 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. 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

${\text{Minimize }}\;f_{0}(x)$ ${\text{subject to: }}\$ $f_{i}(x)\leq 0,i=1,\ldots ,m$ $Ax=b$ where $f_{0},\ldots ,f_{m}$ 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 $x^{*}$ such that $x^{*}$ is strictly feasible (i.e. all constraints are satisfied and the nonlinear constraints are satisfied with strict inequalities).

Mathematically, Slater's condition states that strong duality holds if there exists an $x^{*}\in \operatorname {relint} (D)$ (where relint denotes the relative interior of the convex set $D:=\cap _{i=0}^{m}\operatorname {dom} (f_{i})$ ) such that

$f_{i}(x^{*})<0,i=1,\ldots ,m,$ (the convex, nonlinear constraints)
$Ax^{*}=b.\,$ ## Generalized Inequalities

Given the problem

${\text{Minimize }}\;f_{0}(x)$ ${\text{subject to: }}\$ $f_{i}(x)\leq _{K_{i}}0,i=1,\ldots ,m$ $Ax=b$ where $f_{0}$ is convex and $f_{i}$ is $K_{i}$ -convex for each $i$ . Then Slater's condition says that if there exists an $x^{*}\in \operatorname {relint} (D)$ such that

$f_{i}(x^{*})<_{K_{i}}0,i=1,\ldots ,m$ and
$Ax^{*}=b$ then strong duality holds.