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

Let [math]\displaystyle{ f_1,\ldots,f_m }[/math] be real-valued functions on some subset D of Rⁿ. We say that the functions satisfy the Slater condition if there exists some x in the relative interior of D, for which f_i(x)<0 for all i in 1,...,m. We say that the functions satisfy the relaxed Slater condition if:

• Some k functions (say f₁,...,f_k) are affine;
• There exists some x in the relative interior of D, for which f_i(x)≤0 for all i in 1,...,k, and f_i(x)<0 for all i in k+1,...,m.

Application to convex optimization

Consider the optimization problem

[math]\displaystyle{ \text{Minimize }\; f_0(x) }[/math]

[math]\displaystyle{ \text{subject to: }\ }[/math]

[math]\displaystyle{ f_i(x) \le 0 , i = 1,\ldots,m }[/math]

[math]\displaystyle{ Ax = b }[/math]

where [math]\displaystyle{ f_0,\ldots,f_m }[/math] are convex functions. This is an instance of convex programming. Slater's condition for convex programming states that there exists an [math]\displaystyle{ x^* }[/math] 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 means states that strong duality holds if there exists an [math]\displaystyle{ x^* \in \operatorname{relint}(D) }[/math] (where relint denotes the relative interior of the convex set [math]\displaystyle{ D := \cap_{i = 0}^m \operatorname{dom}(f_i) }[/math]) such that

[math]\displaystyle{ f_i(x^*) \lt 0, i = 1,\ldots,m, }[/math] (the convex, nonlinear constraints)

[math]\displaystyle{ Ax^* = b.\, }[/math]

Generalized Inequalities

Given the problem

[math]\displaystyle{ \text{Minimize }\; f_0(x) }[/math]

[math]\displaystyle{ \text{subject to: }\ }[/math]

[math]\displaystyle{ f_i(x) \le_{K_i} 0 , i = 1,\ldots,m }[/math]

[math]\displaystyle{ Ax = b }[/math]

where [math]\displaystyle{ f_0 }[/math] is convex and [math]\displaystyle{ f_i }[/math] is [math]\displaystyle{ K_i }[/math]-convex for each [math]\displaystyle{ i }[/math]. Then Slater's condition says that if there exists an [math]\displaystyle{ x^* \in \operatorname{relint}(D) }[/math] such that

[math]\displaystyle{ f_i(x^*) \lt _{K_i} 0, i = 1,\ldots,m }[/math] and

[math]\displaystyle{ Ax^* = b }[/math]

then strong duality holds.