  This work addresses data-driven inverse optimization (IO), where the goal is
to estimate unknown parameters in an optimization model from observed decisions
that can be assumed to be optimal or near-optimal solutions to the optimization
problem. The IO problem is commonly formulated as a large-scale bilevel program
that is notoriously difficult to solve. Deviating from traditional exact
solution methods, we propose a derivative-free optimization approach based on
Bayesian optimization, which we call BO4IO, to solve general IO problems. We
treat the IO loss function as a black box and approximate it with a Gaussian
process model. Using the predicted posterior function, an acquisition function
is minimized at each iteration to query new candidate solutions and
sequentially converge to the optimal parameter estimates. The main advantages
of using Bayesian optimization for IO are two-fold: (i) it circumvents the need
of complex reformulations of the bilevel program 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. We apply the proposed method to
three computational case studies, covering different classes of forward
optimization problems ranging from convex nonlinear to nonconvex mixed-integer
nonlinear programs. Our extensive computational results demonstrate the
efficacy and robustness of BO4IO to accurately estimate unknown model
parameters from small and noisy datasets. In addition, the proposed profile
likelihood analysis has proven to be effective in providing good approximations
of the confidence intervals on the parameter estimates and assessing the
identifiability of the unknown parameters.
