Representation of Chance-Constraints With Strong Asymptotic Guarantees

Given ϵ ∈ (0,1), a probability measure μ on Ω ⊂ ℝ
^p
and a semialgebraic set K ⊂ X x Ω, we consider the feasible set X
_∈
* = {x ∈ X : Prob[(x,ω) ∈ K] ≥ 1 - ∈} associated with a chance-constraint. We provide a sequence of outer approximations X
_∈
^d
= {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
\X
_∈
*) → 0 as d increases, where λ is the Lebesgue measure on X. Inner approximations with same guarantees are also obtained.