  We study the problem of learning a linear model to set the reserve price in
an auction, given contextual information, in order to maximize expected revenue
from the seller side. First, we show that it is not possible to solve this
problem in polynomial time unless the \emph{Exponential Time Hypothesis} fails.
Second, we present a strong mixed-integer programming (MIP) formulation for
this problem, which is capable of exactly modeling the nonconvex and
discontinuous expected reward function. Moreover, we show that this MIP
formulation is ideal (i.e. the strongest possible formulation) for the revenue
function of a single impression. Since it can be computationally expensive to
exactly solve the MIP formulation in practice, we also study the performance of
its linear programming (LP) relaxation. Though it may work well in practice, we
show that, unfortunately, in the worst case the optimal objective of the LP
relaxation can be O(number of samples) times larger than the optimal objective
of the true problem. Finally, we present computational results, showcasing that
the MIP formulation, along with its LP relaxation, are able to achieve superior
in- and out-of-sample performance, as compared to state-of-the-art algorithms
on both real and synthetic datasets. More broadly, we believe this work offers
an indication of the strength of optimization methodologies like MIP to exactly
model intrinsic discontinuities in machine learning problems.
