A generalisation of the chance-constrained Charnes-Cooper approach
José A. Díaz-García, Francisco J. Caro-Lopra
TL;DR
The paper generalises the Charnes–Cooper chance-constrained framework to stochastic linear programming by allowing all uncertain parameters to follow elliptically contoured distributions. The core idea is an invariant deterministic reformulation that combines the mean and a risk term via Z(\\mathbf{x}) = k_1 E(z) + k_2 \,\sqrt{Var(z)} with k_1 + k_2 = 1, enabling distribution-invariant solutions across elliptical models. It establishes that key statistics admit t-distribution forms (e.g., T \\sim t_{p-1} or T_i \\sim t_{N_i-1}) irrespective of the specific elliptical member, and provides explicit reformulations for four cases of randomized parameters (c, A, b, and all). The approach relies on sample-based estimators (means and covariances) and yields nonlinear or linear-inequality deterministic programs solvable with standard solvers; a manufacturing example illustrates how different weights on mean versus risk affect the optimal production plan. Overall, the method broadens practical applicability of chance-constrained optimization by accommodating flexible elliptical distributions and data-driven parameter estimation.
Abstract
A generalisation of the Charnes-Cooper chance-constrained approach is proposed in the setting of the family of elliptically contoured distributions. The new relaxed stochastic linear programming is notably invariant under the entire class of probability distributions.