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.^[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.

Formulation

Let ${\displaystyle f_{1},\ldots ,f_{m}}$ be real-valued functions on some subset ${\displaystyle D}$ of ${\displaystyle \mathbb {R} ^{n}}$. We say that the functions satisfy the Slater condition if there exists some ${\displaystyle x}$ in the relative interior of ${\displaystyle D}$, for which ${\displaystyle f_{i}(x)<0}$ for all ${\displaystyle i}$ in ${\displaystyle 1,\ldots ,m}$. We say that the functions satisfy the relaxed Slater condition if:^[3]

• Some ${\displaystyle k}$ functions (say ${\displaystyle f_{1},\ldots ,f_{k}}$) are affine;
• There exists ${\displaystyle x\in \operatorname {relint} D}$ such that ${\displaystyle f_{i}(x)\leq 0}$ for all ${\displaystyle i=1,\ldots ,k}$, and ${\displaystyle f_{i}(x)<0}$ for all ${\displaystyle i=k+1,\ldots ,m}$.

Application to convex optimization

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. Slater's condition for convex programming states that there exists an ${\displaystyle x^{*}}$ 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 ${\displaystyle x^{*}\in \operatorname {relint} (D)}$ (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]

Generalized Inequalities

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]

See also

References