On tractable approximations of randomly perturbed convex constraints

We consider a chance constraint Prob{ξ : A(x, ξ) ∈ K} ≥ 1 - e (x is the decision vector, ξ is a random perturbation, K is a closed convex cone, and A(·,·) is bilinear). While important for many applications in optimization and control, chance constraints typically are "computationally intractable", which makes it necessary to look for their tractable approximations. We present these approximations for the cases when the underlying conic constraint A(x,ξ)∈ K is (a) scalar inequality, or (b) conic quadratic inequality, or (c) linear matrix inequality, and discuss the level of conservativeness of the approximations.