Quantile Inverse Optimization: Improving Stability in Inverse Linear Programming

Inverse linear programming (LP) has received increasing attention because of its potential to infer efficient optimization formulations that can closely replicate the behavior of a complex system. However, inversely inferred parameters and corresponding forward solutions from the existing inverse LP methods can be highly sensitive to noise, errors, and uncertainty in the input data, limiting their applicability in data-driven settings. We introduce the notion of inverse and forward stability in inverse LP and propose a novel inverse LP method that determines a set of objective functions that are stable under data imperfection and generate forward solutions close to the relevant subset of the data. We formulate the inverse model as a large-scale mixed-integer program (MIP) and elucidate its connection to biclique problems, which we exploit to develop efficient algorithms that solve much smaller MIPs instead to construct a solution to the original problem. We numerically evaluate the stability of the proposed method and demonstrate its use in the diet recommendation and transshipment applications.