  Recently, machine learning, particularly message-passing graph neural
networks (MPNNs), has gained traction in enhancing exact optimization
algorithms. For example, MPNNs speed up solving mixed-integer optimization
problems by imitating computational intensive heuristics like strong branching,
which entails solving multiple linear optimization problems (LPs). Despite the
empirical success, the reasons behind MPNNs' effectiveness in emulating linear
optimization remain largely unclear. Here, we show that MPNNs can simulate
standard interior-point methods for LPs, explaining their practical success.
Furthermore, we highlight how MPNNs can serve as a lightweight proxy for
solving LPs, adapting to a given problem instance distribution. Empirically, we
show that MPNNs solve LP relaxations of standard combinatorial optimization
problems close to optimality, often surpassing conventional solvers and
competing approaches in solving time.
