On data-driven chance constraint learning for mixed-integer optimization problems

Abstract

When dealing with optimization problems, decision-makers often face high levels of uncertainty associated with partial information, unknown parameters, or complex relationships between these and the problem decision variables. In this work, we develop a novel Chance Constraint Learning (CCL) methodology with a focus on mixed-integer linear optimization problems, which combines and extends ideas from the literature on chance constraint and constraint learning. While constraint learning aims to model the functional relationship between straight-forward and non-tractable decision variables through the embedding of predictive models within the optimization problem, chance constraints set a probabilistic confidence level for a single or a set of constraints to be fulfilled. One of the main issues when establishing a learned constraint arises when we need to set further bounds for its response variable: the fulfillment of these is directly related to the accuracy of the predictive model and its probabilistic behavior. Therefore, the proposed CCL makes use of piece-wise linearizable machine learning models to estimate conditional quantiles of learned variables, providing a data-driven solution for chance constraints and adding probabilistic guarantees over constraints for learned variables. Open access software has been developed for use by practitioners. Furthermore, the benefits of CCL have been tested in two real-world case studies, demonstrating how robustness is added to optimal solutions when probabilistic bounds are established for learned constraints.