  This work addresses inverse linear optimization where the goal is to infer
the unknown cost vector of a linear program. Specifically, we consider the
data-driven setting in which the available data are noisy observations of
optimal solutions that correspond to different instances of the linear program.
We introduce a new formulation of the problem that, compared to other existing
methods, allows the recovery of a less restrictive and generally more
appropriate admissible set of cost estimates. It can be shown that this inverse
optimization problem yields a finite number of solutions, and we develop an
exact two-phase algorithm to determine all such solutions. Moreover, we propose
an efficient decomposition algorithm to solve large instances of the problem.
The algorithm extends naturally to an online learning environment where it can
be used to provide quick updates of the cost estimate as new data becomes
available over time. For the online setting, we further develop an effective
adaptive sampling strategy that guides the selection of the next samples. The
efficacy of the proposed methods is demonstrated in computational experiments
involving two applications, customer preference learning and cost estimation
for production planning. The results show significant reductions in computation
and sampling efforts.
