# Slater's condition

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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.[3]

## Formulation

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]

## 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]

## References

1. ^ Slater, Morton (1950). Lagrange Multipliers Revisited (PDF). Cowles Commission Discussion Paper No. 403 (Report). Reprinted in Giorgi, Giorgio; Kjeldsen, Tinne Hoff, eds. (2014). Traces and Emergence of Nonlinear Programming. Basel: Birkhäuser. pp. 293–306. ISBN 978-3-0348-0438-7.
2. ^ Takayama, Akira (1985). Mathematical Economics. New York: Cambridge University Press. pp. 66–76. ISBN 0-521-25707-7.
3. ^ Borwein, Jonathan; Lewis, Adrian (2006). Convex Analysis and Nonlinear Optimization: Theory and Examples (2nd ed.). Springer. ISBN 0-387-29570-4.
4. ^ a b Boyd, Stephen; Vandenberghe, Lieven (2004). Convex Optimization (pdf). Cambridge University Press. ISBN 978-0-521-83378-3. Retrieved October 3, 2011.