BO4IO: A Bayesian optimization approach to inverse optimization with uncertainty quantification

Data-driven inverse optimization (IO) aims to estimate unknown parameters in an optimization model from observed decisions. The IO problem is commonly formulated as a large-scale bilevel program that is notoriously difficult to solve. We propose a derivative-free optimization approach based on Bayesian optimization, BO4IO, to solve general IO problems. The main advantages of BO4IO are two-fold: (i) it circumvents the need of complex reformulations or specialized algorithms and can hence enable computational tractability even when the underlying optimization problem is nonconvex or involves discrete variables, and (ii) it allows approximations of the profile likelihood, which provide uncertainty quantification on the IO parameter estimates. Our extensive computational results demonstrate the efficacy and robustness of BO4IO to estimate unknown parameters from small and noisy datasets. In addition, the proposed profile likelihood analysis effectively provides good approximations of the confidence intervals on the parameter estimates and assesses the identifiability of the unknown parameters.