Slater's condition

Formulation

Let f_1,\ldots,f_m be real-valued functions on some subset D of \mathbb{R}^n. 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) for all i in 1,\ldots,m. We say that the functions satisfy the relaxed Slater condition if:

• Some k functions (say f_1,\ldots,f_k) are affine;
• There exists x \in \operatorname{relint} D such that f_i(x) \le 0 for all i=1,\ldots,k, and f_i(x) for all i=k+1,\ldots,m.

Application to convex optimization

Consider the optimization problem : \text{Minimize }; f_0(x) : \text{subject to: }\ :: f_i(x) \le 0 , i = 1,\ldots,m :: Ax = b where 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 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 x^* \in \operatorname{relint}(D) (where relint denotes the relative interior of the convex set D := \cap_{i = 0}^m \operatorname{dom}(f_i)) such that :f_i(x^) (the convex, nonlinear constraints) :Ax^ = b.,

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^) and :Ax^ = b then strong duality holds.

References

References

1. Slater, Morton. (1950). ["Lagrange Multipliers Revisited"](https://cowles.yale.edu/sites/default/files/files/pub/d00/d0080.pdf }} Reprinted in {{cite book). Birkhäuser.
2. Takayama, Akira. (1985). "Mathematical Economics". Cambridge University Press.
3. Nemirovsky and Ben-Tal. (2023). "Optimization III: Convex Optimization".
4. (2004). "Convex Optimization". Cambridge University Press.