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. 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.^[2]

Details

Given the problem

$\text{Minimize }\; f_0(x)$

$\text{subject to: }\$

$f_i(x) \le 0 , i = 1,\ldots,m$

$Ax = b$

with $f_0,\ldots,f_m$ convex (and therefore a convex optimization problem). Then Slater's condition implies that strong duality holds if there exists an $x^* \in \operatorname{relint}(D)$ (where relint is the relative interior and $D = \cap_{i = 0}^m \operatorname{dom}(f_i)$) such that

$f_i(x^*) < 0, i = 1,\ldots,m$ and

$Ax^* = b.\,$^[3]

If the first $k$ constraints, $f_1,\ldots,f_k$ are linear functions, then strong duality holds if there exists an $x^* \in \operatorname{relint}(D)$ such that

$f_i(x^*) \le 0, i = 1,\ldots,k,$

$f_i(x^*) < 0, i = k+1,\ldots,m,$ and

$Ax^* = b.\,$^[3]

Generalized Inequalities

Given the problem

$\text{Minimize }\; f_0(x)$

$\text{subject to: }\$

$f_i(x) \le_{K_i} 0 , i = 1,\ldots,m$

$Ax = b$

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.^[3]

References

1. Slater, Morton (1950). Lagrange Multipliers Revisited (PDF) (Report). Cowles Commission Discussion Paper No. 403.
2. Borwein, Jonathan; Lewis, Adrian (2006). Convex Analysis and Nonlinear Optimization: Theory and Examples (2 ed.). Springer. ISBN 978-0-387-29570-1.
3. Boyd, Stephen; Vandenberghe, Lieven (2004). Convex Optimization (pdf). Cambridge University Press. ISBN 978-0-521-83378-3. Retrieved October 3, 2011.