  Linear programming has been practically solved mainly by simplex and interior
point methods. Compared with the weakly polynomial complexity obtained by the
interior point methods, the existence of strongly polynomial bounds for the
length of the pivot path generated by the simplex methods remains a mystery. In
this paper, we propose two novel pivot experts that leverage both global and
local information of the linear programming instances for the primal simplex
method and show their excellent performance numerically. The experts can be
regarded as a benchmark to evaluate the performance of classical pivot rules,
although they are hard to directly implement. To tackle this challenge, we
employ a graph convolutional neural network model, trained via imitation
learning, to mimic the behavior of the pivot expert. Our pivot rule, learned
empirically, displays a significant advantage over conventional methods in
various linear programming problems, as demonstrated through a series of
rigorous experiments.
