  Inverse linear programming (LP) has received increasing attention due to its
potential to generate 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
method can be highly sensitive to noise, errors, and uncertainty in the input
data, limiting its 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 solutions close to the relevant subset of
the data. We formulate the inverse model as a mixed-integer program and
elucidate its connection to bi-clique problems, which we exploit to develop
efficient heuristics. We also show how this method can be used for online
learning. We numerically evaluate the stability of the proposed method and
demonstrate its practical use in the diet recommendation and transshipment
applications.
