Scenario Approximations of Chance Constraints

We consider an optimization problem of minimization of a linear function subject to the chance constraint Prob{G(x, ξ) ∈ C} ≥ 1 − ε, where C is a convex set, G(x, ξ) is bi-affine mapping and ξ is a vector of random perturbations with known distribution. When C is multi-dimensional and ε is small, like 10−6 or 10−10, this problem is, generically, a problem of minimizing under a nonconvex and difficult to compute constraint and as such is computationally intractable. We investigate the potential of conceptually simple scenario approximation of the chance constraint. That is, approximation of the form G(x, η) ∈ C, j = 1, ..., N , where {η}j=1 is a sample drawn from a properly chosen trial distribution. The emphasis is on the situation where the solution to the approximation should, with probability at least 1− δ, be feasible for the problem of interest, while the sample size N should be polynomial in the size of this problem and in ln(1/ε), ln(1/δ).