  In many applications, when building linear regression models, it is important
to account for the presence of outliers, i.e., corrupted input data points.
Such problems can be formulated as mixed-integer optimization problems
involving cubic terms, each given by the product of a binary variable and a
quadratic term of the continuous variables. Existing approaches in the
literature, typically relying on the linearization of the cubic terms using
big-M constraints, suffer from weak relaxation and poor performance in
practice. In this work we derive stronger second-order conic relaxations that
do not involve big-M constraints. Our computational experiments indicate that
the proposed formulations are several orders-of-magnitude faster than existing
big-M formulations in the literature for this problem.
