  Linear programs with quadratic regularization are attracting renewed interest
due to their applications in optimal transport: unlike entropic regularization,
the squared-norm penalty gives rise to sparse approximations of optimal
transport couplings. It is well known that the solution of a quadratically
regularized linear program over any polytope converges stationarily to the
minimal-norm solution of the linear program when the regularization parameter
tends to zero. However, that result is merely qualitative. Our main result
quantifies the convergence by specifying the exact threshold for the
regularization parameter, after which the regularized solution also solves the
linear program. Moreover, we bound the suboptimality of the regularized
solution before the threshold. These results are complemented by a convergence
rate for the regime of large regularization. We apply our general results to
the setting of optimal transport, where we shed light on how the threshold and
suboptimality depend on the number of data points.
