Data-Driven Chance Constrained and Robust Optimization under Matrix Uncertainty

To solve optimization problems with matrix uncertainty, a novel optimization approach is proposed based on chance-constrained and robust optimization, which focuses on constraints with continuous uncertainty, especially with matrix uncertainty. In chance-constrained approach, constraints with matrix uncertainty are always regarded as joint chance constraints, which can be simplified into individual chance constraints and can be further reformulated into algebraic constraints by robust methods. Motivated by reformulation of chance constraints with right-hand side uncertainty, a novel formulation of constraints with left-hand side uncertainty is proposed, where the uncertainty is described as intervals related to the confidence level of chance constraints. Through using kernel density estimation, confidence sets of uncertain parameters are built to approximate unknown true probability density functions. The approach is illustrated with a motivating and process industry scheduling example with energy consumption uncertainties.