Representation of Chance-Constraints With Strong Asymptotic Guarantees

Abstract

Given e ∈ (0, 1), a probability measure µ on Ω ⊂ Rp and a semi-algebraic set K ⊂ X × Ω, we consider the feasible set X(e) = {x ∈ X : Prob[(x, ω) ∈ K] ≥ 1 − e } associated with a chance-constraint. We provide a sequence outer approximations X_d(e) = {x ∈ X : h_d (x) ≥ 0}, d ∈ N, where h_d is a polynomial of degree d whose vector of coefficients is an optimal solution of a semidefinite program. The size of the latter increases with the degree d. We also obtain the strong and highly desirable asymptotic guarantee that λ(X_d(e) \X (e)) → 0 as d increases, where λ is the Lebesgue measure on X. Finallu inner approximations with same asymptotic guaranties are also obtained by considering the complement.