Slater's condition

Consider the optimization problem

${\displaystyle {\text{Minimize }}\;f_{0}(x)}$

${\displaystyle {\text{subject to: }}\ }$

${\displaystyle f_{i}(x)\leq 0,i=1,\ldots ,m}$

${\displaystyle Ax=b}$

where ${\displaystyle f_{0},\ldots ,f_{m}}$ 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 ${\displaystyle x^{*}}$ such that ${\displaystyle x^{*}}$ 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 ${\displaystyle x^{*}\in \operatorname {relint} (D)}$ (where relint denotes the relative interior of the convex set ${\displaystyle D:=\cap _{i=0}^{m}\operatorname {dom} (f_{i})}$) such that

${\displaystyle f_{i}(x^{*})<0,i=1,\ldots ,m,}$ (the convex, nonlinear constraints)

${\displaystyle Ax^{*}=b.\,}$[4]

Given the problem

${\displaystyle {\text{Minimize }}\;f_{0}(x)}$

${\displaystyle {\text{subject to: }}\ }$

${\displaystyle f_{i}(x)\leq _{K_{i}}0,i=1,\ldots ,m}$

${\displaystyle Ax=b}$

where ${\displaystyle f_{0}}$ is convex and ${\displaystyle f_{i}}$ is ${\displaystyle K_{i}}$-convex for each ${\displaystyle i}$. Then Slater's condition says that if there exists an ${\displaystyle x^{*}\in \operatorname {relint} (D)}$ such that

${\displaystyle f_{i}(x^{*})<_{K_{i}}0,i=1,\ldots ,m}$ and

${\displaystyle Ax^{*}=b}$

then strong duality holds.[4]