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

Exploring the Power of Graph Neural Networks in Solving Linear Optimization Problems