Data-driven individual and joint chance-constrained optimization via kernel smoothing

We propose a data-driven, nonparametric approach to reformulate (conditional) individual and joint chance constraints with right-hand side uncertainty into algebraic constraints. The approach consists of using kernel smoothing to approximate unknown true continuous probability density/distribution functions. Given historical data for continuous univariate or multivariate random variables (uncertain parameters in an optimization model), the inverse cumulative distribution function (quantile function) and the joint cumulative distribution function are estimated for the univariate and multivariate cases, respectively. The approach relies on the construction of a confidence set that contains the unknown true distribution. The distance between the true distribution and its estimate is modeled via ϕ-divergences. We propose a new way of specifying the size of the confidence set (i.e., the ϕ-divergence tolerance) based on point-wise standard errors of the smoothing estimates. The approach is illustrated with a motivating and an industrial production planning problem with uncertain plant production rates.